


\ 






. ^oV* 








<f>^ *■»»«' ^■?, 



.^'\ 





4 o 















<^^^ 



^o 



















^_ ♦»»<.»* ^1^ 



,4 „ » „ -*• 




•J-" . 













"oV^ 





, , . , ^ '' 



0^ c- = ..'^o '^^ 



.V 



' ^0■ 




.0' 



'^>_ ..v 







u^^ 






'J. 



^ ^^ 














,<( 



■^0' 




.0' 



'^*^/'^\/ "°^'^-'/ ^^.'•3^\/ "o^'' 

■'V* aA r<\\ sS/P //1-. . r> *-i ?— -ffl HF—^ "^ \A 






^^ *o.o- ^^ 







XT^' A 






.^^ 










<^ "o ^ 




o^ '^v To s ^ A 



"°. 




c 










ROBINSON'S 

MATHEMATICAL, PHILOSOPHICAL, 

AND ASTEONOMICAL 

CLASS BOOKS, 



PUBLISHED AND FOR SALE BY 



JACOB EHNST, 

No. 112 Main Street, Cincinnati. 



FOR SALE IN" 
boston: 

B. E. MCSSEY ft CO.; P.. DAVIS & CO.: W. J, REYNOLDS & CO.: PHILLIPS, SAMPSOX & CO, 

NEW YORK: 

MASON brothers; D. BL'RGESS & CO.; NEWMAN & IVISON; PRATT, WOODFORD & CO.; 
A. S. BARNES & CO. 

PHILADELPHIA: 

UPPINCOXT, GRAMBO & CO.; THOMAS, COWPERTHWAITE t CO.; 
E. H. BUTLER k CO.; URIAH HUNT & SON. 

BUFFALO, N. Y. 

PHINNEY & CO.; MILLER, ORTON k MULLIGAN. 

SYRACUSE, N. Y. 

E. H. BABCOCK & CO. 
AND IHE PRINCIPAL BOOKSELLERS IN THE UNITED STATES. 



ROBmSOFS CLASS BOOKS, 



BOBINSON'S THEOEETICAL AND PRACTICAL AEITHMETIC. 
KEY TO AEITHMETIC. 
ELEMENTAEY TEEATISE ON ALGEBRA. 
ALGEBRA, UNIVERSITY EDITION. 
KEY TO ALGEBRA. 
GEOMETRY. 

SURVEYING AND NAVIGATION. 
ASTRONOMY, UNIVERSITY EDITION. 
ASTRONOMY. 

CONCISE MATHEMATICAL OPERATIONS. 
NATURAL PHILOSOPHY. 



ROBINSON'S CLASS BOOKS 



ROBINSON'S THEORETICAL AND FBACTICAL ARITHMETIC, 

In which, in addition to the usual modes of operation, the science of numbera, 
the Prussian canceling system, and other important abbreviations hold a 
prominent place. 

KEY TO ROBINSON'S ARITHMETIC. 



ROBINSON'S ELEMENTARY TREATISE ON ALGEBRA, 

For beginners. In this work, some very common euluects are presented in 
a new light. 

From the author's experience and strict attention to the preparation of suit- 
able text-books, the public are assured that they will find this a very desirable 
work for the place it is designed to fill. 

ROBINSON'S ALGEBRA, UNIVERSITY EDITION, 

Being a full course of the sciecce in a clearer and more concise form than any 
other work of the kind heretofore published. It contains all the^ modern im- 
provements, and develops the true spirit of the science, and is highly appreci- 
ated by the most important institutions of learning in the Eastern States, as 
well as in the West. 

# 

KEY TO ROBINSON'S ALGEBRA, 

For the use of teachers, and for those who study without a teacher, contain- 
ing, also, the Indeterminate and Diophantine analysis in concise form. 



ROBINSON'S GEOMETRY, 

Containing Plane and Spherical Trigonometry, Conic Sections, and the necessary 
Logarithmic tables, for practical use. 

This book is designed to give the student a knowledge of Geometry, at once 
theoretical, practical and efficient. The clearest methods of demonstration are 
employed according to the nature of the proposition, whether it be strictly 
Geometrical or Algebraical, or partaking of the combined character and power 
of both. In short, the special attention of all interested in mathematical 
science is called to this work. 



EOBINSON'S SURVEYmG AND NAVIGATION. 

•A Treatise ou Surveying and Navigation, uniting the theoretical, the practical, 
and the educational features of these subjects. This work, in comparison with 
former works on the same subject, is greatly modernized and simplified. It 
contains also many collateral subjects. Its style and manner is such as to 
force itself upon the mind of the learner, 

EOBINSON'S ELEMENTARY ASTRONOMY. 

An abridged edition of the above, 228 pages, in which practical astronomy is 
not included, designed for a classbook in schools. 

ROBINSON'S ASTRONOMY, UNIVERSITY EDITION. 

In this work, facts are not only stated, but the manner of arriving at these 
facts is fully developed. 

The subject of solar eclipses, both general and local, is more simple and 
comprehensively treated than in any former work; and simplicity, conciseness, 
and mathematical philosophy, are its distinctive characteristics. 

Immediately after its publication it was adopted in the Normal Schools of 
Massachusetts, the Albany Academy, and many other leading institutions in 
the East. 

ROBINSON'S ANALYTICAL GEOMETRY, 

AND THE DIFFERENTIAL AND INTEGRAL CALCULUS. 

When a new, a more brief and beautiful form of mathematical investigation is 
discovered, it is natural for its votaries to glorify and mystify it. Such has been 
the case with the subjects in this volume. But no new piinciples have been dis- 
covered — no higher prmciples exist than the definitions and axioms in common 
geometry, — common sense is the highest mathematical law. 

Analytical Geometry is a system of drawing out geometrical truths by the use 
of symbols referred to known lines, called co-ordinates, — and the calculus is 
but an extension of analytical geometry, and the ultimate principles of all are 
the principles of common calculation, (calcu lus.) To show this, and to make 
the whole plain and practical was the great object of this work. Every mathe- 
matical student should have a copy, whether he uses it as a text book or not. 

ROBINSON'S CONCISE MATHEMATICAL OPERATIONS, 

Being a sequel to the Author's Class Books, with miich additional matter. A 
work essentially practical, designed to give the learner a proper appreciation 
of the utility of mathematics; embracing the gems of Science, from common 
Arithmetic through Algebra, Geometry, the Calculus, and Astronomy. 



ROBINSON'S NATURAL PHILOSOPHY, 

228 pages, in which there is more real philosophy than can be found in the 
same number of pages in any other book. Every principle is brought to the 
mind in a clear and practical point of view. This volume contains many philo- 
sophical problems to exercise the learner, and gives him a definite understand- 
ing of the principles of the steam engine, and is the only book which contains 
a full representation of the magnetic telegraph. 



.A 



ELEMENTS 



OF 



ANALYTICAL GEOMETRY, 



AND THE 



DIFFERENTIAL AND INTEGKAL 



C A L C U L U S-. 



BY H. N. ROBINSON, A. M. 

AUTHOR OF A COURSE OF MATHEMATICS, INCLUDING SURVEYING ANO 
NAYIGATION, ASTRONOMY , AND NATURAL PHILOSOPHY. 



FIRST STANDARD EDITION, 




GIN GIN N ATI- 
JACOB ERNST, 112 MAIN STREET, 
1856. 






Entered according to Act of Congress, in the jjrear 1856, 
BY H. N. ROBINSON, tyi^-^ ^if^ 
In the Clerk's Office of the Northern District of New York. 



BTEEEOTYPED BY D. HILLS & CO. 
141 MAIN ST., CINCINNATI. 



PREFACE. 



Setting aside Astronomy and Natural Philosophy the following treatise 
is the sixth volume of a course of mathematics by the same author on the 
same general plan of familiar inductive instruction. All mathematical 
science is positive, sure, and simple, and it is capable of being set forth in 
a natural, clear, and comprehensive light ; and to attain this end all our 
labors have been directed. Hence, we have aimed at a familiar, rather than 
a cold, scientific style, and have embraced every opportunity to give princi- 
ples a practical application, and have in every way made exertions to reach 
the minds of those we hope to instruct. 

We know, however, that our views will not be generally received, — we 
cannot at once convince people that science is truly simple, — they feel that 
whatever is not at once comprehended by them is necessarily vast and com- 
plex, and this is the greatest obstruction to mental progress the mind has to 
encounter. Those who can look at simple nature as she is, will learn with 
great rapidity; others will always be beclouded, and if at times they should 
here and there catch a momentary view of the simplicity of science, that 
very simplicity will only serve to perplex and confound them. 

The impression is abroad that Analytical Geometry and the Differential 
and Integral Calculus are very abstruse and incomprehensible subjects, and 
so they are without other light than that which is furnished by many of the 
text books. Scarcely one of them explains to its readers the objects and the 
aims of its investigations, — they merely direct the learner to do thus and 
so, and he will find this and that result. Not a word of philosophical expla- 
nation — not a word in respect to the object of the pursuit, which would 
enable one to go forward, relying on his own knowledge and strength. 

In this respect we hope to be unlike most others, — we have essayed to 
give to the learner the true object of the investigation before him, and have 
taken every occasion to enlighten his comprehension, requiring him to read 
an equation just as it is, and to give to it the most simple interpretation, and 
not wander away to the ends of the earth in search of intricasies which do 
not exist. 

iii> 



iv. PREFACE. 

Tlaose who have studied our elementary geometry will have less difficulty 
than others in analytical geometry, for that work is half analytical ; trigo- 
nometry in that work is almost purely analytical ; but the absolute analytical 
geometry is in the work before us. Algebra applied to geometry is not ana- 
lytical geometry as at first view some might suppose, for that is only solving 
problems on principles already established ; it is not investigating general 
principles, but applying those already known. 

Analytical geometry is strictly what the term implies; it is a minute and 
careful investigation of a few obvious and well known truths in geometry 
which we combine and compare to discover what other geometrical truths 
inevitably flow from them, the language used being algebraic, with all its 
signs, symbols, and powers of combination. 

Those who are natural algebraists will find very little difficulty in ana- 
lytical geometry; but others, for a time are seriously troubled to interpret 
and comprehend the full import of algebraic equations geometrically applied. 
When the first chapter becomes well understood, there will be no serious 
difficulty in any subsequent part of the subject. When once the equation 
of a straight line in a plane is well understood, the whole theory of analyti- 
cal geometry is before the mind, and the equations of all other lines, whether 
straight or curved, cannot be misunderstood. A person who really under- 
stands the equation of a straipTit line, can readily construct the line from 
the equation, and hence, every teacher should insist on such construc- 
tion as long as he can find the least hesitation in the student to construct 
any equation that maybe offered. According to this idea, we have given 
several practical examples, under various conditions — but as the teacher 
and the pupil can easily propose examples without number, we thought it 
not best to take up space in the book with many mere practical examples. 

Analytical geometry is comparatively a modern science, and a few years 
ago it was not a subject of study even in our colleges, — hence it is that 
many teachers, and others of the old Euclidian school of geometry, do not 
well appreciate, and are in fact prejudiced against it. Let not this discourage 
the young and ambitious student; the modern analysis must now be learned 
by all who have any valid claims to science, nor does it impose on them any 
additional burdens. 

The common simple truths of common geometry, will always be learned 
in the common way, until materials enough are gathered together to use the 
analysis — and then analysis should be used, because it affords the widest 



PREFACE. V. 

field for the exercise of judgment ; it calls into exercise the inventive powers, 
and taxes the memory very little with unimportant particulars. 

" It is in fact the only method by which the student can advance beyond 
the bare rudiments of science without an expense of time and labor wholly 
disproportioned to the ends attained, the only method which gives at once 
a progressive and a self-sustaining power." For these reasons we approxi- 
mated to it as much as possible in our work on common geometry. 

In this we have been as clear and elementary as possible, without diluting 
the subject in the least. In forms we have been as high toned as any other 
author, and in the extent and application of this science we surpass many 
others. We have illustrated every variety of curves, and have taken every 
opportunity to compare algebraic forms to geometrical lines. This will be 
seen in our geometrical solutions of geometrical equations, and in our de- 
lineations of the higher curves. 

The calculus is a branch of analytical geometry, although that term might 
be applied to any thing admitting computation. The differential calculus 
takes into consideration small differences. The differential of a quantity is 
the difference between two quantities of the like kind, when one is very 
nearly equal to the other, and from this definition alone, the ingenious stu- 
dent might find the differential for himself. 

In geometry, we can conceive a line to be formed by the motion of a point, 
a surface to be formed by the motion of a line, and a solid to be formed bj 
the motion of a plane, either moving parallel with itself or by revolving 
about an axis. 

Thus, when a point moved and formed a line, the point was said to Jloie, 
and a small amount of such motion was called by the English mathemati- 
cian the fluxion of the line, — a very small surface formed by the flowing 
of a line which bounded any side of a surface, was called i\i& fluxion of that 
surface, and so on. The fluxions of the English mathematicians is the same 
thing as the differential of the French — and of late all have adopted the 
differential technicalities of the Fiench. 

The differential calculus is generally regarded as a very abstruse and dif- 
ficult science, but this is the fault of the text books used ; when that science 
is really comprehended, it is found to be no more abstruse than any other of 
the mathematical sciences ; indeed, it is but an extension of algebra and 
geometry, using no new and no other system of computation. 

Any branch of science, however simple, would be perfectly dark and ab- 



vi. PREFACE. 

struse to us, provided we have no prior and proper apprehension of the object 
of pursuit, and this our text books have never given in respect to the diifer- 
ential and integral calculus. Authors on this subject have contented them- 
selves \7ith saying that in the investigations will be found two kinds of 
quantities, variables and constants ; they then define what symbols denote 
the variables and what denote the constants, and then direct the student 
what to do. 

"We have taken great pains to remedy this deficiency, and we shall feel 
much disappointment if our efforts are pronounced abortive on these points. 

Notwithstanding the great importance we attach to the illustrations of 
what the differential calculus is, they occupy but very little space, but two 
or three pages at the most, and they are chiefly to be found in the intro- 
duction. 

The differential calculus may be applied to any thing susceptible of 
change. For instance, every one knows that the variation of the length of 
the shadow of any object on any fixed plane must correspond to the varia- 
tion of the sun's altitude, and the variation of altitude depends on the latitude 
of tlie plane, the declination of the sun, and the time of day. In short, the 
differential or small change in the shadow of an object compared with the 
object itself, must have a corresponding variation in the time of day,* and 
any scientific computation between two such small corresponding differences 
is one application of the differential calculus. Thus, the differential calculus 
is the ratio of small corresponding differences. 

The integral calculus is the converse of the differential, somewhat as the 
cube root is the converse operation of cubing the root. It is more difficult to 
find the cube root of a number than it is to cube the root, but still the one 
operation is the converse of the other, and the one is not directly obvious 
from the other. 

In many cases the integral calculus is more difficult than its correspond- 
ing differential, but it is not so in every case. In shorty if the student will 
look at nature in its true and simple light, all difficulties will quickly 
vanish, and his progress in science become pleasant and invaluable. 

* See 43d miscellaneous Example, near the end of the volume. 



CONTENTS. Tii. 

ANALYTICAL GEOMETRY. 

SECTION I. CHAPTER I. 

To find the equation of any straight line in a plane, 9 12 

On the intersection of straight lines, 19 21 

Teansfoemation of co-ordinates, 24 26 

Polar co-ordinates, 26 27 

CHAPTER II. LINES OF THE SECOND ORDER. 

The equation of the circle, — polar equation, &c 27 37 

Application of the polar equation, 37 41 

CHAPTER III. CONIC SECTIONS. 

The Ellipse, its equation and area, 42 55 

Conjugate diameters, 55 64 

CHAPTER IV. THE PARABOLA. 

The equation of the curve, its properties, <fec 66 70 

Oblique co-ordinates, 70 

The property of projectiles,. 72 

The application of the polar equation of the parabola to the solu- 
tion of common quadratic equations in algebra, 76 -78 

CHAPTER V. THE HYPERBOLA. 

The equation of the curve, 80 

The various properties of the hyperbola, 82 86 

Change of co-ordinates, and properties disclosed thereby, 90 92 

Asymptotes of the hyperbola, 92 96 

Conclusion, and general results, 100 

SECTION II. CHAPTER I. 

On the geometrical representation of equations, 101 110 

Particular examples given, 106 110 

CHAPTER II. 

Curve lines corresponding to equations, 110 112 

Algebraic equations of the third and fourth degree, solved geo- 
metrically by a parabola and circles, Ill 115 

Curves not of the conic sections, 115 120 

CHAPTER III. 

Straight lines in space, 120 128 

To find the inclination of any line in space to the three axes of 

co-ordinates, • 125 

CHAPTER IV. 

On the equation of a plane, 128 140 

To find the intersection of two planes, 134 

Examples to illustrate principles taught in Sec. II, 138 140 

THE DIFFERENTIAL CALCULUS, 

SECTION I. CHAPTER I. 

Definitions and illustrations, .141 1*43 

Rules to differentiate simple variables, 144 150 

Rule to find the differential coefficient of any quantity under a radical, 151 

Examples under the foregoing rules, 153 156 

CHAPTER II. 
On the differential of circular functions,. 157— — 163 

CHAPTER III. 

On the differential of exponential and logarithmic quantities,.. .163 171 

.Rule to differentiate a logarithmic variable, 164 

Examples to illustrate the principles taught in this chapter,. . . .167 171 

CHAPTER IV. 

On successive differentials, 172 176 

Taylor's Theorem — illustrated and demonstrated, 172 -176 



viii. CONTENTS. 

Maclaurin's Theorem illustrated, 176— —182 

The MODULUS of the common system of logarithms, numerically 

determined, 182 

Remarks on Maclaurin's and on Taylor's theorems, 186 190 

CHAPTER V. 

The development of functions containing two or more variables, . 190 197 

The result of the development applied, 196 

CHAPTER VI. 

Application of the calculus to plane curves, 197 201 

Application of the truths taught in this chapter, 199 200 

SECTION II. CHAPTER I. 

Maxima and minima — general principles, 201 ^203 

Application of maxima and minima to various problems, 203 216 

CHAPTER II. 

Differential coefficients applied to curves, *. .216 230 

Points of inflection, .222 ^224 

Insulated points, .228 230 

CHAPTER III. 

Osculating curves, — evolutes and involutes, 230 237 

CHAPTER IV. 

The differential expression for polar curves, 237 239 

CHAPTER V. 

On transcendental curves, 239 250 

Other geometrical differentials, 249 250 

THE INTEGRAL CALCULUS. 

CHAPTER I. 

Differentials and integrals compared, 251 ^263 

Arbitrary constants determined, 253 ^254 

Examples in integration, ^263 

CHAPTER II. 
The integration of circular differentials, 263 ^266 

CHAPTER III. 

Integration by series, 266 ^269 

CHAPTER IV. 
Integration of rational fractions, 269 276 

CHAPTER V. 

Integration by parts, — Formulas for integration, 276 286 

Formulas applied to examples, 281 287 

CHAPTER VI. 

Integration of irrational fractions, — logarithmic integrals, 287 294 

CHAPTER VII. 

Integration of exponential differentials, — series of John Bernoulli, 294 300 

Integration by parts, and the series of John Bernoulli compared, . 299 305 

A new logarithmic series, 302 303 

CHAPTER VIII. 

The integration of circular differentials of multiple arcs, 305 309 

CHAPTER IX. 

Successive Integrations, 309 313 

CHAPTER X, XI, XII. 

Geometrical integrals, 313 333 

CHAPTER XIII. 

On the integration of homogeneous and linear differentials, 334 339 

Miscellaneous examples 339 348 



ANALYTICAL GEOMETRY. 



Introductory Remarks. 



Geometry, purely analytical, is the investigation of general 
geometrical truths by the aid of algebraic equations. 

Many of the demonstrations in our elementary geometry, 
trigonometry, and conic sections, where algebra is used, are 
partially analytical, not purely so. 

Algebra applied to geometry must not be mistaken for ana- 
lytical geometry, because the operator in either case uses the 
same mathematical motive power, the science of algebra. 

Algebra applied to geometry only contemplates the use of 
algebra in solving problems — and all the geometrical truths are 
supposed to be previously known. 

Analytical geometry draws out algebraically, all the necessary 
results from given data. 

To pursue this branch of science successfully, the student 
must perfectly comprehend the nature and import of algebraic 
expressions — must understand general proportion, and the com- 
mon rules of plane trigonometry. 

We shall adopt the same general notation as other writers on 
this subject. 

With this brief introduction, we commence 

CHAPTER I. 

PROPOSITION I.—PROBLEM. 
To find the equation of a straight line. 

We now propose to show the equation which can be made to 
represent any straight line that can be drawn in a plane, and 

.9 



10 



ANALYTICAL GEOMETRY. 



without a perfect comprehension of this, in both letter and spirit, 
it will be useless for the pupil to advance. 

Draw a vertical and a horizontal line. On these two lines all 
measurements are to be made. The point of intersection of 
these two lines we shall call the zero point. Horizontal measures 
to the right from this point we shall call pluSy to the left minus. 
Vertical measures from the zero point upwards we call plus, 
downwards minus. 

It ha& become the custom of all writers to denote unknown 
and indefinite distance&along the horizontal by x^ and along the 
vertical by y. 

Hence the horizontal line of measure itself is called the axis 
of X, and the vcBtical line the axis of y, and they are marked 
as in the figure. 

The point A in the adjoining fig- 
ure is the zero point. Draw any line 
as L'L through this point, and de- 
signate the natural tangent of the 
angle LAX by a, (the radius being 
unity.) 

Then take any distance on AX as 
AF, and represent it by x, and the 
perpendicular distance PJSf put equal 
to yr. 
Then by trigonometry we have 

Rad. : tan. MAP i :; AF : PM 
\\.a'.\.x\y ox y.7=.ax (1) 

Now this- equation is general ; that is, it applies to any point 
M on the line AL, because we can make aj greater or less, and 
PJf will be greater or less in like proportion, and J!f will move 
along on the line AL as we move P on the line AX. Because 
the point JSfwill continue on the line -4Athrough all changes of a? 
and y, we say that y=ax, is the equation of the line AL. 

Now let us diminish x to 0, and the equation reduces to y=0 
in the same time, which brings MonAo the point A. 

Let x pass the line YY\ it then, becomes — x. AP' and the 




STRAIGHT LINES. U 

con esponding value of y will be P'M\ and being below the line 
X'X will therefore be minus. 

Therefore ±y=±aiP 

is the general equation of the line LL', extending indefinitely in 
either direction. . 

If the tangent a becomes less, the line will incline more to- 
wards the line X'X. When a=0 the line will coincide with 
X'Xf when infinite, it will coincide with YV 

Now let AF"' be +x, and a become —a, then P"'M'" will 
correspond to y, and becomes mimis y, because it is below the 
axis XX'. Or, algebraically y= — ax, indicating some point M'" 
below the horizontal axis. 

Now we think it has been shown that y=ax may represent any 
line asZZ' passing through A from the \st into the Zd quadrant, 
and y= — ax may be made to represent any line as UL" passing 
through A from the 2d into the 4th quadrant. 

Therefore y=do.ax 

may he vnade to represent any straight line passing through the zero 
point. 

In case we have — a and — x, that is, both a and x minus at 
the same time, their product will be -\-ax, showing that y must 
be plus by the rules of algebra. 

We now request the learner to examine these geometrical lines 
and see whether they correspond. 

When we have — a we must draw the line from A to the right 
and below AX; then XAL" is the angle whose natural tangent is 
— a. But the opposite angle X'AL" is the same in value. 

When we have — x we must take the distance as AP" to the 
left of the axis YY', and the corresponding line F'M" is above 
XX', and therefore ^Zw5, as it ought to be. 

But the equation of a straight line passing through the zero 
point is not sufficiently general for practical application; we will 
therefore suppose a line to pass in any direction across the axis 
YY', cutting it at the distance AB or AB (dbJ) or h distance 



J2 



ANALYTICAL GEOMETRY. 



I 



above or below the zero point A, 
and find its equation. 

Through the zero point A draw a 
line ^ A" parallel to ML. 

Take any point on the line AX 
and through P draw PM parallel to 
-41^ then ABMN ^Y\\\ be a parallelo- 
gram. 

Put AP=x. PM=y. The tan- 
gent of the angle N'AP=a. Then 
will KP=ax. 

To each of these equals add NM=^b, then we shall hare 

y=:zax-\-b 

for the algebraic expression corresponding to the point M, and 
as J!f is any variable point on the line ML corresponding to the 
variations of x, this equation is said to be the equation of the 
line ML. 

When b is minus the line is then QL', and cuts the axis YF' 
in D, a point as far beloAv ^ as ^ is above A. 

Hence we perceive that the equation 

may represent the equation of any line in the plane TAX. 

If we give to «, x, and b, their proper signs, in each case of 
application we may write 

. y=:a:c-\-b 

for the equation of any straight line i7i a plane. 

To fix in the minds of learners a complete comprehension of 
the equation of a straight line, we give the following practical 



EXAMPLES. 



1. Draw the line whose equation is y=2x-\-S. (1) 
Then draw the line represented by y= — rr-|-2 (2) 

and determine where these two lines intersect. 




STRAIGHT LINES. 13 

Draw YY" and XX' at right 
angles, and take any conveni- 
ent space for the unit of mea- 
sure, as 1, 2, 3, (fee. 

Equation (1) is true for all 
v^alues of x and y. It is true 
then when a:=0. But when 
a;=0 the point on the line must 
be on the axis YY' , 

When ir=0. y=3. 

This shows that the line sought for must cut YY' at the point 

+3. 

The equation is equally true when y=0. But when y=0, 
the corresponding point on the line sought must be on the 
axis XX' y and on the same supposition the equation becomes 

0=237+3, Or x=—\\. 

That is, midway between — 1 and — 2 is another point in the 
line which is represented by y=2ar-(-3, but two points in any 
right line must define the line : therefore ML is the line sought. 

Taking equation (2) and making a; =0 will give 2/=2, and 
making 2/=0 will give a;=2 : therefore MQ must be the line 
whose equation is y= — a:-|-2, and these two lines with the axis 
XX''form the triangle LMQ, whose base is 3-| and altitude about 2-^. 

But let the equations decide, (not about,) but exactly the posi- 
tion of the point M oi intersection. 

This point being in both lines, the co-ordinates x and y cor- 
responding to this point are the same, therefore we may subtract 
one equation from the other, and the result will be a true equa- 
tion, giving 

3:r+l=0. Or x=—^. 

Eliminating x from the two equations we find 2/=2j-. 

2. For another example we require the projection of the line 
represented by the equation 

y= -—2. 

420 

Making x=0, then y= — 2. Making y=0, then ir= — 840. 
2 



14 ANALYTICAL GEOMETRY. 

Using the last figure, we perceive that the line sought for must 
pass through S two units below the zero point, and it must take 
such a direction SVsiS to meet the axis XX' at the distance of 
840 units to the left of zero. Hence its absolute projection is 
practically impossible. 

3. Construct the line whose equation is y=z9.x-\-5. 

4. Construct the line whose equation is y= — 3x — 3. 

PHOPOSITION II.— PROBLEM. 

To find the distance between two given points in the plane of 
the co-ordinate axes. Also, to find the angle made by the line 
joining the two given points, and the axis of X. 

Definition. — A point is said to be given when its co-ordinates 
are known. Known co-ordinates are designated by x\ y', — x" 
y" — x"'y y'" ; which are read x prime, x" second, &c. 

When the point designated by the co-ordinates i^ no particular 
one, we write simply x and y, to represent its co-ordinates. 

Let the two given points be P 
and Q, and because the point P 
is said to be given, we know the 
two distances -4i\^and NP. 
AN=x\ NP=y\ 

And because the point Q is 
given we know the two distances 

AM=x" and MQ=:y'\ 

AM—AN'=:NM=:PR=x"—x\ 

MQ—ME= QE=y"—y\ 

In the right angled triangle PJRQ we have 

{PPy+(PQy=(PQ)». Put J)=PQ. 

That is, D''=(x"—x'y+(y"—y'y, 




Or D=^(x"—xy+(y"-y)K 

Thus we discover that the distance between any two given 
points is equal to the square root of the sum of the squares of th$ 
difference of their abscisses and ordinaies. 



STRAIGHT LINES. 16 

If one of these points be the origin or zero point, then «'=0 
or y'=0, and we have 

a result obviously true. 

To find the angle between PQ and AX. 

PR is drawn parallel to AX, therefore the angle sought is the 
same in value as the angle QPR. 

Designate the tangent of this angle by a, then by trigonometry 

we have 

PR \ RQ \\ radius : tan. QPR. 

That is, x" — x : y"—y' ::!:«. 

Whence a= ^ -. 

x" — x' 

In case y"=^y', PQ will coincide with PR, and be parallel to 

AX, and the tangent of the angle will then be 0, and this is 

shown by the equation, for then 

«=_JL_=o. 

x" x' 

Incase x"=.x', then PQ will coincide with RQ and be paral- 
lel to A Y, and tangent a will be infinite, and this too the equa- 
tion shows, for if we make x"=^x' or x" — .t''=0, the equation will 
become 


PROPOSITION III. 
To find the equation of a line drawn through a given point. 
Let P be the given point : The equation must be in the form 

y—ax-{-b. ( 1 ) 

Because the line must pass through the given point whose 
co-ordinates are x' and y\ we must have 

y'=ax'+b. (2) 

Subtracting (2) from (1) we have 

y—y'=a{x—x') (3) 

for the equation sought. 

In this equation a is indefinite, as it ought to be, because an 



16 ANALYTICAL CxEOMETRY. 

infinite number of straight lines can be drawn througb the 
point P. 

We may give to y' and x' their numerical values, and give any 
value whatever to a, then we can construct a particular line that 
will run through the given point P. 

Suppose a;'=2, ?/'==3, and make a=4. 
Then the equation will become 

y— 3=4(ar— 2). 
Or 3/= 4a? — 5. 

This equation is obviously that of a straight line, hence (3) 
IS of the required form. 

PROPOSITION lY. 

To find the equation of a line which passes through two given 
points. 

More definitely, we say find the 
equation of the line which passes 
through the two given points P 
and Q. 

As the equation is to be that 
of a line, it must correspond to 
y=ax-\-h. (1) 
As it must pass through the 
given point P, whose co-ordinates 
are x and y, we must have 
y^=ax'-\-h. (2) 
Subtracting (2) from (1) we have 

y—y'=.a{x—x'). (3) 

Because the line must also pass through the other point Q, we 
must have (Prop. II.) 

a=^ — ^^. 

x" X ' 

Substituting this value of a in (3) we have 

\x X / 

the equation sought. 




STRAIGHT LINES. 



17 



PROPOSITION V. 

To find the equation of a straight line which shall pass through 
a given point and make a given angle with a given line. 

The equation of the given line must be in the form 

y^ax-\-h. (1) 

Because the other line must pass through a given point its 
equation must be (Prop. III.) 

y—y'^a'{x—x). (2) 

We have now to determine the value of a. 

When a and a are equal, the two lines must be parallel, and 
the inclination of the two lines will be greater or less according 
to the relative values of a and a. 

Let PQhQ the given line (the 
tangent of its angle with the axis 
of X equal a) and PR the other 
line which shall pass through 
the given point P and make a 
given angle QPR. The tangent 
of the angle PRX=:a'. 

Because PRX=PQR-{-QPR. 

QPR=^PRX—PQR. 

Tan. QPR=tan.(PRX—PQR.) 

As the angle QPR is supposed to be known or given, we may 
put m to designate its tangent, and m is a known quantity. 
Now by trigonometry we have 

a' — a 




m=t&n.(PRX—PQR) = 



}-\-aa' 



(3) 



Whence 



a = L — 



This value of a' put in (2) gives 

g—g=(lL^\(x—x') 
\ 1 — 7na / 

for the equation sought. 



(4) 



18 ANALYTICAL GEOMETRY. 

Corollary 1 . When the given inclination is 90°, m its tangent 
is infinite, and then «'== — -. We decide this in the following 



manner : 

An infinite quantity cannot be increased, therefore on that 
a-^m -L m 



supposition 



becomes — or 

-ma — ma 



Application^. — To make sure that we comprehend this propo- 
sition and its resulting equation, we give the following example : 
The equation of a given line is y=2:c-4-6. 

Draw another line that will in- 
tersect this at an angle of 45° and 
pass through a given point P, 
whose co-ordinates are 

x=^, 2/'= 2. 
Draw the line J/IV corresponding 
to the equation y=^x-\-6. Locate 
the point jP from its given co-or- 
dinates. 

Because the angle of intersection 
is to be 45°, m=l, a=2. 

Substituting these values in (4) we have 
y~2=~3(x—3^). 
Or 7/=—3x-{-12i. 

Constructing the line MB correSj.onding with this equation, 
we perceive it must pass through F and make the angle iOfi? 
45°, as was required. 

The teacher can propose any number of like examples. 

Corollary 2. Equation (3) shows the tangent of the angle 
of the inclination of any two lines whose tangents are a and a. 
That is, we have in o-eneral terms 




\-\-aa 
In case the two lines are parallel m=0. Whence «'=«, an 
obvious result. 



STRAIGHT LINES. 19 

In case the two lines are perpendicular to each other, m must 
be infinite, and therefore we must put 

l-f-aa'=0 

to correspond with this hypothesis, and this gives 

a 
a result found in Cor. 1 . 

To show the practical value of this equation we require the 
angle of inclination of the two lines represented by the equations 
yr=^x — 6, and y= — x-{-9,. 

Here a=3 and a'= — 1. Whence 
4 
1—3 

This is the natural tangent of the angle sought, and if we 
have not a table of natural tangents at hand, we will take the 
log. of the number and add 10 to the index, then we shall have 
in the present example 10.301030 for the log. tangent which 
corresponds to 63° 26' 6" nearly. 

The minm sign merely indicates the position of the angle, it 
is hdow the angular point. 

PROPOSITIOJ^ VI. 

To find the co-ordinates which will locate the point of intersec- 
tion of two straight lines given by their equations. 

We have already done this in a particular example in Prop. 1, 
and now we propose to show general expressions for the same 
thing. 

Let yz=ax-{-b be the first line. 

And y=ax-\-h' be the second line. 

At their point of intersection y and x in both equations will 
represent the same point. 

Therefore we may subtract one equation from the other, and 
the result will be a true equation. 

For the sake of perspicuty, let x^ and y^ represent the co- 



20 ANALYTICAL GEOMETRY. 

ordinates of the point of intersection in each line, then by sub- 
traction (a — a)x^-\-(b — 6'}=0 

Whence x^= — S / and y,= — ~ — . 

(a — a) a — a 

EXAMPLE. 

At what point will the two following lines intersect : 

3/=— 2^+1. 
And y=5x-\-10. 

Here a=—2, a'=5, h=zl, 5'=10. Whence ar=— f , y=—2}. 
If we take another line not parallel to either of these, the 
three will form a triangle. 

Then if we locate the three points of intersection and join them, 
we shall have the triangle. 

PROPOSITIOI^- VII. 

To draw a perpendicular from a given point to a given straight 
line and to find its length. 

Let y=ax-\-h be the equation of the given straight line, and 
x'j y', the co-ordinates of the given point. 

The equation of the line which passes through the given point 
must take the form 

y — y'=a'[x — x). (Prop. III.) 

And as this must be perpendicular to the given line, we must 

have a'= — -. Therefore the equations for the two lines must be 
a 

y=ax-\-h for the given line. (1) 

And y — y'= — -[x — x') for the perpendicular line. 

Or y= — _ x-\- i--\-y' \ for the perpendicular. (2) 

Let x^ and y^ represent the co-ordinates of the point of inter- 
section of these two lines. Then by Prop. VI, 

—a \a / 



• 1 X f 

h——y 
a 



andy^; 



a+\ ) 1+a 

a a 



STRAIGHT LINES. 21 

^ / ah — x' — mj \ •, h-\-ax-{-a^y' 

Or ^i=— ( :rTT-^)' and 2/, = -3L_T:_:^ 

Or we may conceive x and y to represent the co-ordinates of 
the point of intersection, and eliminating y from (1) and (2) we 
shall find x as above, and afterwards we can eliminate y. 

Now the length of the perpendicular is represented by 



Whence ^[{^E^t^tz^^y +^ = the 

perpendicular. 

If we put u=^h-\-ax' — y', the quantities under the radical, will 
become 









Whence the perpendicular ==b— I E ' ^ . 

EXAMPLES. 

1. The equation of a given line is y=3x — 10, and the co- 
ordinates of a given point are .t'=4 and y'=5. 

What is the length of the perpendicular from this given point 
to the given straight line? ^^5^ _t_. 789^ 

2. The equation of a line is y= — 5x — 15, and the co-ordinates 
of a given point are x'=:4 and y'=5. 

What is the length of the perpendicular from the given point 
to the straight line? Ans. 7.84-j-. 

PROPOSITION^' YIII. 

To find the equation of a straight line which will Used the 
angle contained hy the inclination of two other straight lines. 
Let yz=^ax-\-h ( 1 ) 

And y=^ax-\'h' (2) 

be the equations of two straight lines which intersect, and the 
co-ordinates of the point of intersection are 

.,=-(^) y.=«-:^ (Prop. VI.) 
\a — a / a — a 



22 



ANALYTICAL GEOMETRY. 



We now require a third line which shall pass through the 
same point of intersection and form an angle with the axis of 
X (the tangent of which may be represented by m) which will 
bisect the angle made by the inclination of the other two lines. 
Whence by (Prop. V.) the equation of the line sought must be 



y—y 



i{x-^x^) 



(3) 



in case we can find the value of m. 




Let PN represent the line 
corresponding to equation (1)» 
PM the line whose equation is 
(2), and PR the line required. 

Now the position or inclina- 
tion of PN to AX depends en- 
tirely on the value of a, and the 
inclination of Pilf depends on a\ 
and all are entirely independent 
of the position of the point P, 

Now RPN=RPX'—NPX and MPR=MPX^RPX'. 

Whence by the application of a well known equation in plane 
trigonometry, (Equation (29), p. 143, in Robinson's Geometry,) 
we have 

m — a 



tan. RPX=tan.{RPX'^XPX): 



1-^am 



And tan. MPR=tan.{MPX—RPX')=- 



l-\-am 

But by hypothesis these two angles i^PA^and MPR are to be 
equal to each other. Therefore 

m — a a! — m 



Whence 



1 -}-awi 1 -\-am 
2 , 2fl— aa') , 
a-\-a 



(4) 



This equation will give two values of m ; one will correspond 
to the line PR, the other will be its supplement. 

If the proper value of m be taken from this equation and put 
in (3); then (3) will be the equation required. 



MiHiiHIMMiiMiMMMIHii 



STRAIGHT LINES. 23 

Practically we had better let the equations stand as they are, 
and substitute the values of a, a! x, and y, corresponding to any 
particular case. 

To illustrate the foregoing proposition we propose the following 

EXAMPLES. 

Two lines intersect each other : 

y= — 2x-\-5 is the equation of one line. (1) 
And i/=4z-]-6 is that of the other line. (2) 

Find the equation of the line which bisects the angle con- 
tained by these two lines : 

Here a= — 2, a'=4, b=z5, 5'=6. 

Whence x^= — }, and y, = L6. 
Thus far (3) becomes 

And (4) becomes 

7n^ -|-9m= 1 . 
Whence m=0.1093 or wz=— 9.1095. 

y-V=0.i095(^+-J). (3) 

Or y_y^_9 1095(a:+i). (4) 

Equation (4) is the line required; (3) is the line at right 
angles to the line required. All will be obvious if we construct 
lines (1), (2), (3), and (4). 

For another example, find the equation of a line which bisects 
the angle contained by the two lines whose equations are 
y=x-}-12, y=—20x-\-2. 



Observatiox. — Two straight lines whose equations are 
y=ax-]-b and y=zax-\~b' 
will always intersect at a point (unless a=a) and with the axis 
of Fform a triangle. The area of such a triangle is expressed 

'' -(S)xC^^). 



JN 



ANALYTICAL GEOMETRY. 



Transformation of Co-ordinates. 




Let A be the zero point of the primitive system, and A' the 
zero point of this new system. 

Let AD^=x and DM=y. Also, 
let A'B=x and BM=y, we are to 
find the equation connecting a; to a: 
and y to y. The change of position 
from A to A' must be given in all 
cases. 

Make AP=a and PA'^h, It 
is now visible that x=a-\-x' and 
y=h+y'. 

We may transform the origin 
from Aio A2 ox to A^ or to ^4, as well as to A', by giving to 
a and b their proper corresponding values and signs. 

PROPOSITION IX. 

To find formulas for passing from a system of rectangular to 
a system of ohliqm co-ordinates from a different origin. 

Let AB=^a, BA'=b, AP—x, 
PM^y, A'P'=x, P'M^y the 
angle X'AX"=^rR, and the angle 
Y'AX"=-n. Xow by trigonome- 
try we have 

^'^=a;'cos.m KP'=^LH=^x' sm.m 

P'H=^KL^y' cos. n 

And MH^^^y' sin. 71. 

Whence x=a-\-x' coq, w-j-y'cos. n, y=ih-\-x &m. m-f-y'sin. n 
the formulas required. 

Scholium. In case the two systems have the same origin, we 
merely suppress a and h, and then the formulas required are 

x=x' Go?>. m-\-y' Gos. n y=x's,m. 7w-|-y'sin. n. 




STRAIGHT LINES. 26 

PROPOSITION^ X. 

To find the formulas for passing from a system of oblique co- 
ordinates to a system of rectangular co-ordinates, the origin being 
the same. 

Take the formulas of the last problem 

x=x coB.m-{-y' Qos.n, y—x' sm.m-\-y ^m,n. 

We now regard the oblique as the primitive axes, and require 
the corresponding values on the rectangular axes. That is, we 
require the values of x and y'. If we multiply the first by 
sin. %, and the second by cos. w, and subtract their products, y' 
will be eliminated, and if x' be eliminated in a similar manner, 
we shall obtain 

, xsm.n — ycos.w , ycos.m— a;sin. m 

X = ; y =- ; . 

sin.(?i — m) sin. (92 — m) 

Scholium. If the zero point be changed at the same time in 
reference to the oblique system, we shall have 

xsin.n — ycos.n , ^ , yco^.m — x^m.m 



X =za-\- — —-j~ — r — y =H 



sin.(w — m) sin. ( 72 — m) 

We close this subject by the following 



EXAMPLE. 



The equation of a line referred to rectangular co-ordinates is 

y=:axA-y . 

Change it to a system of oblique co-ordinates having the sam® 
zero point. 

Substituting for x and y their values as above, we have 

a;' sin. m-\-y' sin. n=^a{x' cos. w^-|-y'cos. n)-\-b\ 
Reducing 

, (a'cos.m — sin.m)a;' , b' 



sm. n — a cos. m sm. n — a cos. m 



26 



ANALYTICAL GEOMETRY. 



Polar, €o-ordioates. 



When a ]ine is conceived to revolve round a point, that point 
is called a fole, and any other point in such a line referred to 
co-ordinates, is denominated the system of polar co-ordinates. 

Conceive the line AB to revolve 
round the point A a-B a pole. Let 
AB=r. It may be a variable dis- 
tance, and it is then called the raditis 
vector. 

Put the variable angle BAD^v, 
AD=x, DB=y, then by trigo- 
nometry 

x=r COS. V, and y=r sin. v. 




Now from the first of these we have 



cos.v 



(v may re- 



volve all the way round the pole), and as x and cos.v are both 
positive and both negative at the same time, that is, both posi- 
tive in the first and fourth quadrants, and both negative in the 
second and third quadrants, therefore r will always be positive. 

Consequently, should a negative radius appear in any equation, 
we mtist infer that some incompatible conditions have been ad- 
mitted into the equation. 

Scholium 1. If we change the origin now from A to A', 
writing x' and y' for the corresponding co-ordinates, we shall 
have 

x'=a-\-r COS. V 

y'=b-^r sin. V. 

Scholium 2. If in place of estimating the variable angle 
from the line AD the axis, we estimate it from the line Alf 
which makes with the axis the given angle £[AI)=^m, we shall 
have 

x'=a-{-rG08.(v-\-m). 

y'=b-\-r sin. (v-\~m). 



THE CIRCLE. 27 

CHAPTER II. 
liines of the second order. 

Straight lines can be represented by equations of the first 
degree, and they are therefore called lines of the first order. 
The circumference of a circle, and all the conic sections, are 
lines of the second order, because any point in them referred to 
co-ordinates requires equations of the second degree. 

PROPOSITION I. 
7h find the equation of the circle. 

Let the origin be the center of 
the circle. Draw AM\.o any point 
in the circumference, and let fall 
MP perpendicular to the axis of X. 
Put AP=x, PM=:y and AM=P. 

Then the right angled triangle 
^PJf gives 

x^+y'=P^ (1) 

and this is the equation of the circle 
when the zero point is the center. 

When y=:0, x^ = R^, or ±:x=P, that is, Pis at X or A\ 
Whenar=0, y''=R'^, or ±y=R, showing that if on the cir- 
cumference is then at Yov Y'\ 

When X is positive, then P is on the right of the axis of Y, 
and when negative, P is on the left of that axis, or between A 
and A\ 

When we make radius unity, as we often do in trigonometry, 
then x'^-\-y^ = \, and then giving to x oy y any value plus or 
minus within the limit of unity, the equation will give us the 
corresponding value of the other letter. 

In trigonometry y is called the sine of the arc XM, and x its 
cosine. 

Hence in trigonometry we have sin.2-|-cos.^ = 3. 




2a ANALYTICAL GEOMETRY. 

Now if we remove the origin to A' and call the distance 
A'F=x, then AF=x — E, and the triangle AFH gives 

Whence y^=2JRx — x^. (2) 

This is the equation of the circle, when the origin is on the 
circumference. 

When x=0 y=0 at the same time. When x is greater than 
2M, y becomes imaginary, showing that such an hypothesis is in- 
consistent with the existence of the circle. 

There is still a more general equation of the circle when the 
zero point is neither at the center nor in the circumference. 
The figure in the margin will fully illustrate. 

Let AB=c,^ BC==b. Put 
AP=x, or AP'=z'j:, and FMov 
F'M"'==y, CM, CM', &c. each 

In the circle we observe four 
equal right angled triangles. 
The numerical expression is the 
same for each. Signs only indi- 
cate positions. 

Now in case CDM is the tri- 
angle wefx upon, 
We put AP=x, then BF=CD={x—c), 
FM=y, MI)=y—CB=(y—h]. 




Whence 



(x-cy+(y-iy=ji-- 



(1) 



In case CDM' is the triangle, we put AF—x and FM'=y. 



Then 



(x~cy+{b-yy=F'^ 



(2) 



In case CDM'" is the triangle, we put AF'=x, FM"'=y. 

Then (c—xy-{-(y—by=E^ (3) 

If CD'M" is the triangle, we put F'M"^y. 

Then (^c—xY-^{b—yY^R'- (4) 

Equations (1), (2), (3), and (4), are in all respects numeri- 



* We do not take a, because a, in this science is generally understood to 
represent the tangent of an angle. 



THE CIRCLE. 29 

cally the same in value, for (c — x)^=(x — c)^, and {b — y)^ = 
^y — hy. Hence we may take equation (IJ to represent the 
general equation of the circle referred to rectangular co- 
ordinates. 

The equation {x—cY-\-{y—h)'^=zR^ (1) 

includes all the others by attributing proper values and signs to 
c and b. 

If we suppose both c and h equal 0, it transfers the zero point 
to the center of the circle, and the equation becomes 

To find where the circle cuts the axis of X we must make 
y=0. This reduces the general equation (1) to 

Or (^x—cY=:R'^—hK 

Now if h is numerically greater than B, the first member being 
a square, (and therefore positive,) must be equal to a negative 
quantity, which is impossible, — showing that in that case the 
circle does not meet or cut the axis of X, and this is obvious in 
the figure. 

In case 5=i2 then {x — c)^=0, or a?=c, showing that the 
circle would then touch the axis of Xat the point J5. 

To show where the circle cuts the axis of Y, make a;=0 : then 
(1) becomes 

Or (2,_5)2=i22__c2 

This equation shows that if c is greater than iJ, the circle 
does not cut the axis of Y, and this is also obvious from the 
figure. 

If c be less than jB, the second member is positive in value, 



and y=^h±JR^-^c\ 

showing that if it cut the axis at all, it must be in two points, 
as at M\ M'". 
3 



30 ANALYTICAL GEOMETRY. 

PROPOSITION II. 

The supplementary chords in the circle are perpendicular to each 

'dher. 

Definition. — Two lines drawn through the two extremities of 
any diameter of a curve, and which intersect the curve in the 
same point, are called supplementary chords. 

That is, the chord of an arc, and the chord of its supplement. 
In common geometry this proposition is enunciated thus : 
All angles in a semicircle are right angles. 

The equation of a straight 
line which will pass through the 
given point J?, must be of the 
form (Prop. III. Chap. I.) 

y—y'=a{x—x). (1) 

The equation of a straight 
line which will pass through the 
given point X, must be of the 
form y — y'=a\x — x'). (2) 

At the point By y'=0, and a:'= — R, or — x=E. There- 
fore (1) becomes 

y=a(x+F.). (3) 

And for like reason (2) becomes 

y=a\x-E). (4) 

When these two lines intersect, y in (3) is the same as y in 
(4), and x in (3) is the same as x in (4), therefore these equa- 
tions/or the point of intersection may be regarded as two numeri- 
cal equalities ; hence we may multiply them together and obtain 
a true numerical equation, that is, 

y^=aa(x^—Il^). (5) 

But as the point of intersection must be on the curve, by hy- 
pothesis, therefore, x and y must conform to the following equa- 
tion : 

y^+x'^^E^ Or y^ =—l(x^—E^). (6) 

Whence aa'= — 1, or aa'-}-l=0. 




THE CIRCLE. 



31 



This last equation shows that the two lines are perpendicular 
to each other, as proved by (Cor. 2, Prop. V, Chap. I.)* 

Because a and a are indefinite, we conclude that an infinite 
number of supplemental chords may be drawn in the semicircle, 
which is obviously true. 

Scholium. As BDX is a right angled triangle, and BX its 
hypotenuse, it follows that the diameter is greater than any chord. 
As one chord increases, its supplementary chord decreases. 

From the center A let fall the perpendiculars AH, AF. Then 
the two triangles XAH and XBD are equiangular and similar ; 
therefore, as A is the middle point of XB, His the middle point 
of XD, and F is the middle point of BD. AH=^(BD), and 
AF=^[XD). That is, the distance of any chord frmn the center 
is equal to half its supplementary chord. 

PROPOSITION III. 

To find the equation of a straight line which shall be tangent 
to the circumference of a circle. 

Draw a line cutting the curve in 
any two points, as P and Q. De- 
signate the co-ordinates of the point 
P by x\ y\ and of the point Q by 
x", y", and of any other point in the 
line as H by x, y. 

Now the equation of any line 
passing through point H may be 
expressed by 

y=ax-\-h. ( I ) 

* This condition of the perpendicularity of the two 
lines may be more satisfactory to some when they read 
the more direct demonstration. 

Let AB be one line, and AT) another line at right 
angles to it. Join BD, and from A draw AX perpen- 
dicular to BD, and conceive ^A'the axis. The tangent 
of the angle BAX=a, and XAD=^—a\ AX=l, and 
it is the mean proportion between a and -—a'. There- 
fore 

a : 1 : : 1 : —a'. 

Whence — aa'=l or O(i'-fl=0. Q. B, D 





m ANALYTICAL GEOMETRY. 

If the same line passes througli the point F, the equation for 
that point must be 

y'=:ax'-{-b. (2) 

And the same line passing through Q, the equation for that 
point must be 

y"=ax"-}-h. (3) 

Subtracting (3) from (2) and we find 

y'—y"=a(x—x") (4) 

for the equation which passes through the two points P and Q* 

Subtracting (2) from (1) and we have 

y-^y'z=a(x-^x') (5) 

for the equation of the line which passes through the two points 
P and If. 

The line which passes through the three points Q, P, and ff^ 
is expressed in the two equations (4) and (5). 

Conceive the line QPJI to revolve on the point P, so as to 
make Q coincide with P, then the line will be a tangent at P. 

We have now to determine the value of a, when the line becomes 
a tangent at P. 

Because the two points P and Q are in the circumference, we 
have 

Subtracting and factoring the remainder, gives us 

(a:'+^")(a;'-:.")+(/+3/")(/-/')=0. (6) 

The value of (y' — y") taken from (4) and substituted in (6), 
and then divided by (x' — x") will reduce (6) to 
x'+x"+a(y'+y")=0. 

Whence o=— ("^HflV (7) 

This equation is true, however far or near P and Q may bo 
from each other, provided they be on the curve ; and when QPJI 
becomes a tangent at P, x'=x" and y'=y", then (7) reduces to 

i a=-^. (8) 



THE CIRCLE. 33 

This value of a substituted in (5) gives 

y-y=-<(^-^'). (9) 

y 

This is the general equation of a tangent line ; x\ y\ are 
the co-ordinates of the tangent point, and x, y, the co-ordinates 
of any other point in the line. 

Scholium. For the point in which 
the tangent line cuts the axis of X, we 
make y'=0, then 

x=:^=:AT. 

x 

For the point in which it meets the 
axis of Y, we make x'=0, and 

y=~^=AQ. 

y 

Definitions. — A line is said to be normal to a curve when it 
is perpendicular to the tangent line at the point of contact. 

Join APy and if APT is a right angle, then AP is a normal, 
and AB, a portion of the axis of X under it, is called the sub- 
normal. The line BT under the tangent is called the suhtangerU. 

Let us now discover whether APT\^ or is not a right angle. 
Equation (8) shows us the tangent value of the inclination 
of the line PT with the axis of X. 

Put a'= the tangent of the angle PAT, then by trigonometry 






«'=^. 






x' 




But 


x' 


Eq. (8) 


Whence 


aa=L — 1. 


Or a'=— i, 



Therefore AP is at right angles to PT, (Prop. V. Chap. I.) 



ii ANALYTICAL GEOMETRY. 

PROPOSITION IV. 

To find the equation of a line which shall pass through a given 
point without the circle. 

Let H be the given point, and x' and y' its given co-ordinates, 
and X and y the co-ordinates of the tangent point P. 

The equation of the line passing through the two points H 
and P, must be of the form 

y—y'=a{x—x'). (1) 

And if PH is tangent at the point P, and x and y the co- 
ordinates of the point P, equation (8) of the last proposition 
gives us 

x 

— -• 

This value of a put in ( 1 ) and we have 

y—y =— - {x—x ) 
y 

for the equation sought. 

This equation combined with that of the circle 
x^+y'^^R^ 
will determine the values of x and y, and as there will be two 
values to each, numerically equal, it shows that two equal tan- 
gents can be drawn from H, or from any point without the circle, 
which is obviously true. 

Scholium. We can find the value of the tangent FT by 
means of the similar triangles ABP, PBT, which give 
X \ R \ \ y \ FT. 

ft=rI. 

X 

More general and elegant formulas will be found in the cal- 
culus for the normals, subnormals^ tangents, and suhtangents 
applicable to all the conic sections. 

Note to Propositions III and IV of this Chapter. — In the investiga- 
tion of these propositions we followed in the footsteps of others, only hoping 
to be more definite and clear. But were we only in pursuit of results, we 
would have been more brief and practical. 




THE CIRCLE. ' 35 

In these propositions it is not assumed that the radius of the circle is at 
right angles to its tangent when drawn from the center to the point of con- 
tact, but we see no propriety in excluding this geometrical truth so well 
known in elementary geometry, especially when we consider that we have 
all along used the symbol a to represent the tangent of angles on the admis- 
sion that the tangent of an angle was a line drawn at right angles to ihr 
radius from the extremity of the radius. 

Using this truth we would not draw a line 
cutting the curve in two points, but would 
draw the tangent line PT at once, and adinit 
that the angle APT was a right angle. Then 
it is clear that the angle APB= the angle 
PTB. 

Now to find the equation of the line, we let 
x' and y' represent the co-ordinates of the point 
P, and X and y the general co-ordinates of the 

line, and a the tangent of its angle with the axis of X, then by (Prop. Ill, 
Chap. I,) we have 

y' — y=a(x'—x). 

Now the triangle APB gives us the following expression for the tangent 
of the angle APB, or its equal PTB, 

a=-^2 

y 

This value of a put in the preceding equation, will give us 
y'—y—~~~(x'—x). 

y 

Or 2/' 2 — yy'= — x'^-^xx'. 

Whence yy'-\-xx'=R^ the same as before. 



Of the Polar Equation of the Circle. 

The polar equation of a curve is the equation for any point in 
the curve estimated from any fixed point called a pole. The 
variable distance from the pole to any point in the curve is 
called the radius vector, and the angle which the radius vector 
makes with a given straight line is called the variable angle. 

PROPOSITION V. 
To find the polar equation of the circle. 

When the center is the pole or the fixed point, the equation is 
x-'+y^=^B^ (1) 

and the radius vector R is then constant. 



ANALYTICAL GEOMETRY. 




Now let P be the pole, and the 
co-ordinates of that point a and b. 
PM=r, and MPX'=v the variable 
angle. ANz=x and NM=y. Then 
it is obvious that 
x-=a-\-r COS. v, and y±=J-(~**sin. v. 
These values of x and y substi- 
tuted in (1), (observing that cos.^v 
-j-sin. ^a;=l,) will give 

which is the polar equation sought. 

Scholium 1. P may be at any 
point on the plane. Suppose it at^'. 
Then a— — R and ^=0. Substitu- 
ting these values in the equation, 
and it reduces to 

r^ — 9.R cos. vr=^0. 

As there is no absolute term, r=0 
will satisfy one point in the curve, 
and this is true, as P is supposed to be in the curve. Dividing 
by r, and 

r=2i2cos. v. 

This value of r will be positive while cos. v is positive, and 
negative when cos.^ is negative ; but r being a radius vector can 
never be negative, and the figure shows this, as r never passes 
to the left of B, but runs into zero at that point. 

When 2;== 0, 008.^=1, then r=^^'. When z^=90, cos,'y=0, 
and r becomes at B^ and the variations of v. from to 90, de- 
termine all the points in the semicircle BDB' . 

Scholium 2. If the pole be placed at B , then «=-)-i? and 
5=0, which reduces the general equation to 
r= — Si^cos. V. 

Here it is necessary that cos.?; should be negative to make r 
positive, therefore v must commence at 90° and vary to 270° ; 
that is, be on the left of the axis of Indrawn through B\ and 
this corresponds with the figure. 




THE CIRCLE. 37 

Application. — The polar equation of the circle in its most 
general form is 

r^-\-2{acos.v-{'bsm.v)r+a^+b^=E^. (1) 

If we make 5=0, it puts the polar point somewhere on the 
axis of X, and reduces the equation to 

r^+ga cos. v.r+a2=i23. (2) 

Now if we make v=0, then 
will cos. v=l, and the lines 
represented by ±r would refer to 
the points X, X', in the circle. 

This hypothesis reduces the last 
equation to 

r^+2ar=(E^—a=') (3) 
and this equation is the same in 
form as the common quadratic in 
algebra, or in the same form as 

x'^±px=q. 
Whence x—r, ^a=d=:p, and jf^^ — a^=q. 




These results show us that if we describe a circle with the 



radius Jq-^-lp'^ , and place F on the axis of X at a distance 
from the center equal to ^p, then PX represents one value of 
X, and FX' the other. That is, 



^=-lP+j9+lP'=J'^' 



Or x^=^\p—Jx-\-ip^^=:FX\ 

and this is the common solution. 

When p is negative, the polar point is laid off to the left from 
the center at F\ 

The operation refers to the right angled triangle AFM, 



JF=±p, FM=^q, and AM=Jq-{-\pK 
Let the form of the quadratic be 

X^zhpX: 



^;3 



ANALYTICAL GEOMETRY. 



Then comparing this with the polar equation of the circle, we 
have 




Take AX=R and describe a 
semicircle. Take AP=\p and 
AP'=—\p. From P and P' 
draw the lines PM, and P'M' to 
touch the circle. Join AM, AM'. 
Here AP is the hypotenuse of 
a right angled triangle. In the 
first case AP was a side. 

In this figure as in the other, PM=Jq ; but here it is inclined 
to the axis of X; in the first figure it was perpendicular to it. 

The figure thus drawn, we have PX for one value of x, and 
PX' is the other, which may be determined geometrically. 
If x^-\-px= — q 



x=—^p+J\p^—q^PX, or x=—^p--J\p'—q=PX\ 

Observe that the first part of the value of ic, is minus, corres- 
ponding to left position from P. 

If ic^ — px= — q, 

we take P' for one extremity of the line x. 



x=^p+Jlp^—q=P'X, or x=lp-^J\p—q=PX\ 
Here the first part of the value of x, (^p), is plus, because to 
the right of the point P' . 

Because R=J\p^ — q, R or -<4il/' becomes less and less as 
the numerical value of q approaches the value of \p^ . When 
these two are equal, i?=0, and the circle becomes a point. 
When q is greater than \p^ , the circle has more than vanished, 
giving no real existence to any of these lines, and the values of 
x are said to be imaginary. 



We have found another method of geomefrising quadratic equa- 
tions, which we consider well worthy of notice, although it is 
of no practical utility. 




THE CIRCLE. 39 

It will be remembered that the equation of a straight line 
passing through the origin of co-ordinates is 

y=ax, (1) 

and that the general equation of the circle is 

{xd^cY+{y±:hY=RK (2) 

If we make 6=0, the center of the circle must be somewhere 
on the axis of X. 

Let AM represent a line, the 
equation of which is y=^ax, and 
if we take a=l, AM vfiW incline 
46° from either axis, as repre- 
sented in the figure. Hence 
y=x, and making 5=0, these 
two values substituted in (2), 
and that equation reduced, we 
shall find 

This equation has the common quadratic foryn. 

Equation (1) responds to any point in the straight line M'M. 
Equation (2) responds to any point in the circumference BMM\ 

Equations (1) and (2) combined must respond equally to the 
straight line and to the circle. Therefore equation (3) must 
respond to the points M and M', the points in which the circle 
cuts the line. 

That is, PM and PM' are the two roots of equation (3), and 
when one is above the axis of X, as in this figure, it is the posi- 
tive root, and P'M' being below the axis of X, it is the negative 
root. 

When both roots of equation (3) are positive, the circle will 
cut the line in two points above the axis of X. When the two 
roots are mintis, the circle will cut the line in two points below 
the axis of X. 

When the two roots of any equation in the form of (3) are 
equal and positive, the circle will touch the line above the axis 
of X If the roots are equal and negative, the circle will touch 
the line below the axis of X. In case the roots of (3) are im- 
aginary, the circle will not meet the line. 



4^ ANALYTICAL GEOMETRY. 

We give the following examples for illustration : 

To determine the values of y by a geometrical construction of 
this kind, we must make 

c= — 2, and =5. 

2 

Whence i?=3.74, the radius of the circle. Take any distance 
on the axes for the unit of measure, and set off the distance c on 
the axis of X from the origin, for the center of the circle ; — 
to the right, if c is negative, and to the left, if c is positive. 

Then from the center, with a radius equal to R=j2p-\-c^f 
describe a circle cutting the line drawn midway between the two 
axes, as in the fignire. 

In this example the center of the circle is at C, the distance of 
two units from the origin A, to the right. Then, with the radius 
3.74 we described the circle, cutting the line in M and M\ and 
we find by measure (when the construction is accurate) that 
jl/P=3.44, the positive root, and M'P'=^ — 1.44, the negative 
root. 

For another example we require the roots of the follovjing equa- 
tion hy construction: 

y2+6y=27. 

N. B. When the numerals are too large in any equation for 
convenience, we can always reduce them in the following manner: 
Put y=nz, then the equation becomes 

^i5 22_|_g^2=27. 

n, 2 I 6 27 

Or z^-\--z — 



n n 



Now let w= any number what- 
ever. If n=S, then 
z^-\'2z=3, 

Herec=2. tl ^=3. 

2 

Whence E=JlO=3A6, 

At the distance of two units 




CONIC SECTIONS. 41 

to tlie left of the origin, is the center of the circle. We see by 
the figure that 1 is the positive root, and — 3 the negative root. 

But y=w5?, n=3, 2=1, y=3 or — 9. 

We give one more example. 

Construct the equation 

Here c=4, and -'^""^''=—6. Whence i2=2. 
2 

Using the same figure as before, the center of the circle to 
this example is at D, and as the radius is only 2, the circum- 
ference does not cut the line M'M, showing that the equation 
has no real roots. 

We have said that this method of finding the roots of a quad- 
ratic vras of little practical value. The reason of this conclu* 
sion is based on the fact that it requires more labor to obtain 
the value of the radius of the circle than it does to find the 
roots themselves. 

Nevertheless this method is interesting and instructive as an 
algebraic geometrical problem. 

When we find the polar equation of the parabola, we shall then 
have another method of constructing the roots of quadratics which 
vnll not require the extraction of the square root. 



CHAPTER IIL 
Conic Sections. 



If we cut a cone by a plane through its vertex, the section 
will be a triangle. If we cut it by a plane at right angles with 
the axis of the cone, the section thus cut will be a circle. If 
we cut it on one side by a plane parallel to the other side, the 
section will be a parabola. If we cut it by a plane less inclined 
to the base than the sides of the cone, the section will be an 
ellipse. If by a plane more inclined to the base than to the sides 
of the cone, the section will be an hyperbola. 



# ANALYTICAL GEOMETRY. 

Hence, the triangle and the circle might be included in conic 
sections, — but custom has limited the term to the three curves, 
the Ellipse, the Parabola, and the Hyperbola. 

We can and do examine the properties of a triangle and a 
circle without the least regard to a cone whatever. So also, can 
the cone be entirely dispensed with in discussing the properties 
of the ellipse, the parabola, and the hyperbola, and we shall dis- 
pense with it, commencing with 

The Ellipse. 

Definition 1. — An ellipse is a plane curve, confined by two 
fixed points, and the sum of the distances from any point in the 
curve to the fixed points, is constant. 

2. — The two fixed points are called the foci. 

3. — The center is midway in a straight line between the two 
foci. 

4. — A diameter is a straight line passing through the center. 

6. — The major axis is a diameter passing through the foci. 

8. — The minor axis is at right angles to the major axis, passing 
through the center. 

9. — The distance between the center and either /ocms, is called 
the eccentricity when the semimajor axis is unity. 

10. — The parameter of an ellipse is the double ordinate passing 
through one of the foci. 

PROPOSITIOIf I. 

To find the equation of the curve, the origin of the co-<yrdinaies 
beinff in the center, the major axis being given, also the distance 
of the foci from the center. 

The curve in the margin repre- 
sents an ellipse. 

Put CF=c, CA=:A. 
Take any point, as F, and let fall the 
perpendicular Ft. 

By our conventional notation, put 

Ci=x, tF=y. 




THE ELLIPSE. 43 

As F'P-\-PF=<2,A, we may put F'P=A+z, and PF=A—z. 
Then the two right angled triangles F'Pt, FPt, give us 
{c+xY+y^^={A^zY (1) 

{c—xY+y^-={A—zy (2) 

For the points in the curve which cause t to fall between c and 
F, we would have 

{x-cY+y-=.{A-zY (3) 

But when expanded, there is no difference between (2) and 
(3), and by giving proper values and signs to x and y, equations 
(1) and (2) will respond to any point in the curve as well as to 
the point P. 

Substracting (2) from (1), and dividing by 4, we find 

cx=Azy or 2= (4) 

A 

This last equation shows that F'P, the radius vector, varies 

as the abscissa x. 

Add (1) and (2), and divide the sum by 2, and we have 

Substituting the value of z"^ from (4), and clearing of frac- 
tions, we have 

c^A^+A-x^-^A^y'^z^A^+c^xK 
Or A''y^-\-{A^—c^)x^=A-'(A^—c^). (5) 

Now conceive the point P to move along describing the curve, 
and when it comes to the point B, so that D C makes a right angle 
with the axis of Z", the two triangles DCF SLJid. DCF' are right 
angled and equal. DF and DF' each is equal to A, and as 
CF, CF\ each is equal to c, we have 
DC^^A^—c'. 
It is customary to denote Z> C half the minor axis of the ellipse 
by B, as well as half the major axis by -4, and adhering to this 
notation 

^2^^2_c2^ (6) 

Substituting this in (5) we have for the equation of the ellipse 

^2y2^^2^2^^2^2^ 

referred to its center for the origin of co-ordinates. 



44 ANALYTICAL GEOMETRY. 

If we wish to transfer the origin of co-ordinates from the cen- 
ter of the ellipse to the extremity A' of its major axis, we must 
put 

x= — A-\-x\ and y=y'. 

Substituting these values of x and y in the last equation, and 
reducing, we have 

y'^=?^j2Ax'—x'^). 
Or without the primes, we have 

for the equation of the ellipse when the origin is at the extremity 
of the major axis. 

Corollary 1 . If it were possible for J5 to equal A, then c 
must equal 0, as shown by (6). Or, while c has a value, it is 
impossible for JB to equal A. 

If B=A, then c=0, and the equation becomes 
A^y''+A^x''=A^AK 
Or y^-{-x^=:A\ 

the equation of the circle. Therefore the circle may be called 
an ellipse, whose eccentricity is zero, or whose eccentricity is ««/?- 
nitely small. 

Corollary 2. To find where the curve cuts the axis of X, 
make y=0 in the equation, then 

x=±A, 
showing that it extends to equal distances from the center. 

To find where the curve cuts the axis of Y, make x=0, and 
then 

Plus B refers to the point D, — B indicates the point directly 
opposite to £>, on the lower side of the axis of X. 

Finally, let x equal any value whatever less than A, then 
y=.dt^(A^^x^)i. 



THE ELLIPSE. 46 

An equation showing two values of y, numerically equal, indi- 
cating that the curve is symmetrical in respect to the axis of X. 
If we give to y any value less than B, the general equation 
gives 

Showing that the curve is symmetrical in respect to the axis 
of T, 

Scholium. The ordinate which passes through one of the foci, 
corresponds to a;=c. But ^^ — B^=c^. Hence -4^ — c^ or 

A* — x^=£^. Or (A^ — x^)'^=B, and this value substituted 

B^ 2^2 

in the last equation, gives y=dz Whence is the 

A A 

measure of the parameter of any ellipse, by Def. 10. 

PROPOSITION II. 

Every diameter is bisected in the center. 

Let X, and y, be the co-ordinates of the point D, and x\ y\ 
the co-ordinates of the point D'. 
Then by the equation of the curve 

And ^2y'2+^2^'2=^2^3. 

The equation of a line passing 
through the center, must be of the 
form y=ax. 

This equation combined with the equations of the curve, 

gives 

AB aAB 

5^= 




Ja^A^+B^ Ja'^A^+B- 

AB , aAB 



Ja^A'-+B^ Ja^A^+B^ 

These equations show that the co-ordinates of the point Dy 
are the same as those of the point D\ except opposite in signs. 
Hence UB' is bisected at the center. 
4 




46 ANALYTICAL GEOMETRY. 

PROPOSITION III. 

The squares of the ordinates are to one another as the rectangles 
of their corresjmnding abscissas. 

Let y be any ordinate, and x its corres- 
ponding abscissa. Then, by the first 
proposition, we shall have 

Let y' be any other ordinate, and x' its 
corresponding abscissa, and by the same proposition we must 
have 

y'^=:^"{^A^x')x\ 

Dividing one of these equations by the other, omitting com- 
mon factors in the numerator and denominator of the second 
member of the new equation, we shall have 
2/2 _{^A — x)x 
Y^ {^A~x)x'' 

Hence y^ : y'^=(2A—x)x : {9.A—x')x'. 

By simply inspecting the figure, we cannot fail to perceive 
that (9,A — x)y and x, are the abscissas corresponding to the or- 
dinate y^ and (9. A — x'), and x', are the two corresponding to y\ 
Therefore, the squares of the ordinates, &c. Q. E. D. 

Scholium. Suppose one of these ordinates, as y, to represent 
half the minor axis, that is, y'=B. Then the corresponding value 
of «' will be A, and (2A — x') will be A, also. Whence the last 
proportion will become 

y^ : B^=(2A—x)x : AK 

In respect to the third term we perceive that if A'H is repre- 
sented by X, -4^ will be {2 A — x), and if 6"^ is a point in the 
circle, whose diameter is AA, and GHi\iQ ordinate, then 

and the proportion becomes 

r : B'^^'QH.'^ : A^. 
Or y : GE=zB : A. 

Or .4 : B^QH I y^BR 



THE ELLIPSE. 



47 




PROPOSITION IV. 

The area of an ellipse is the mean proportional between the 
areas of two circles, ike diameter of one being the major axis, and 
the diameter of the other, the minor axis. 

Conceive GIT to be a practical as well 
as a mathematical line; or rather, conceive 
it be a ver^ narrow parallelogrom. 

Conceive also other lines Q-'H\ Q"H", 
<fec, drawn so as to fill the whole space 
occupied by the semicircle and semi- 
ellipse.* 

Then by scholium to Prop. Ill, we have 
A : B=:Gff : DH, 

z=G'H' : D'lr. 
= G"H" : D"H\ 

&G. &G. 

But as the sums of proportionals have the same ratio as thel? 
like parts, (see proportion in algebra,) therefore 

A : B :: (Gff+G'JI'+&c.) : {DH-\-D'H'+&,q,) 

But the sum of all the narrow parallelograms represented by 
(^^-|-6'^'^'-j-&c.) is the area of the semicircle on ^'^ : and the 
sum of all the parallelograms represented by {DH-\'D'II'-\-&,g,^ 
is the area of the semi-ellipse. 

But wholes are in the same proportion as their halves, whence 
A : jB=area circle : area ellipse. 
But the area of the circle on the major axis, is rtA^ . 
Substituting this, and the proportion becomes 
A : B=irtA^ : area ellipse. 

Or area ellipse=rt^^, 

which is the mean proportional between {jtA^) and (ttB^ ,) the 
expressions for the areas of the circles, one on the major axis, 
the other on the minor axis. Q. E. D. 



* These narrow parallelograms are called differentials, in the differentia] 
calculus — and the sum of them is called the integral, in the integral cal 
cidus. 



48 



ANALYTICAL GEOMETRY. 




ScHDUUM. Hence the common rule in mensuration to find 
the area of an ellipse. 

Rule — Multiply the semi-major and semi-minor axes together t arid 
multiply that product iy 3. 1 4 1 6. 

PROPOSITIOlf V. 

To find the product of the tangents of two supplementary chords 
with the axis of X. 

Let X, y, be the co-ordinates of 
any point, as P, and x', y', the co- 
ordinates of the point A'. 

Then the equation of a line 
which passes through the two 
points A' and P, ( Prop. Ill, Chap. 
I,) will be 

y—y'—a{x^x). (1) 

Th^ equation of the line which passes through the points A 
and Pf will be of the form 

y-y"=a\x—x"). (2) 

For the given point A\ we have y'=0, and x'= — A, 
Whence (1) becomes 

y=a{xJrA). (3) 

For the given point A we have y"=0, and x"=Ay which 
values substituted in (2) give 

y=a{x—A), (4) 

As y and x are the co-ordinates of the same point P in both 
lines, we may combine (3) and (4) in any manner we please. 
Multiplying them, we have 



y^ =aa'{x^—A^ ). 



(5) 



Because P is a point in the ellipse, the equation of the curve 
gives 



(A'- 



J^—i^^-A^h (6) 



Comparing (5) and (6) we find 

aa'= — -~ for the equation sought. 



THE ELLIPSE. 49 

Scholium 1. In case the ellipse becomes a circle, that is, 
in case A=B, aa'-|-l=0, showing that the angle A'FA would 
then be a right angle, as it ought to be, by (Prop. II, Chap. II,) 

D2 

Because is less than unii^, or aa less than 1,* or radius; 

A^ 

the two angles FA' A and FAA' are together less than 90° ; 

therefore the angle at F is obtuse, or greater than 90°. 

Scholium 2. Since aa has a constant value, the sum of the 
two, a, a , will be least when a=a'. 

Hence the angle at F will be greatest when F is at the vertex 
of the minor axis, and the supplementary chords equal ; and the 
angle at F will become nearer a right angle as F approaches A 
or A' . 

PROPOSITION VX. 

To find the equation of a straight line which shall be tangerU 
to an ellipse. 

Let X, y, be the co-ordinates of 
any indefinite point R, in a line 
cutting an ellipse ; x, y, the co- 
ordinates of the point F, and x", 
y", the co-ordinates of the point 
Q. Also, let a be the tangent of the angle of inclination of the 
line FR with the axis of X. The object is to find the value of 
a when FR is tangent to the ellipse. 

The equation of a line which passes through two points, as 
R and F, must be of the form 

y—y'=a{x—x). (1) (Prop. Ill, Chap. I.) 

The equation for the same line passing through the two points 
R and Q, must be • 

y—y"=a{x~x"). (2) 

And the equation for the same line passing through the two 
points F and Q, must be 
y'—y"=a{x'—x"Y (3) 

*In trigonometry we learn that tan. z cot. j:=i22=l. That is, the pro- 
duct of two tangents the sura of whose arc is 90°, equals 1. When the 
Bum is less than 90°, the product will be a fraction. 




50 



ANALYTICAL GEOMETRY. 



Because the points P and Q are in the curve, the co-ordinates 
of those points must correspond to the following equations : 

By subtraction A^7/'^—f^)-{'B^{x'^—x"^)=0. 

Or A^y'+f )(y'-f )=--£' (x'+x")ix'^x"). (4) 

Dividing (4) by (3) we have 



AHy'+f)=---(x+x"), 



(5) 



IS'ow conceive the line to revolve on the point F until Q co- 
inoides with F, then FF will be tangent to the curve. But 
when Q coincides with F, we shall have 
y'=y" and x=x". 

Whence (5) becomes 

2A''y'=-^—x\ 
a 



Or 



B^x' 



A^y' 

This value of a put in ( 1 ) gives 
B^x' 



y—y 



{x—x'). 



A^y' 

Reducing A''yy'-\-B^xx'=^A''y'^-\-B''x'^ . 

Or A^yy'+B^xx'=A^BK 

This is the equation sought, x and y being the general co- 
ordinates of the line. 

Scholium 1. To find where the tangent meets the axis of X 
we must make y=0. 

This gives x=:^'^-=CT. 

X 

In case the ellipse becomes a cir- 
cle, £—A, and then the equation 
will become 

yy'-\-xx'=A^, 
the equation for a tangent line to a 




THE ELLIPSE. 61 

circle; and to find where this tangent meets the axis of X, we 
make ^=0, and 

a?= = CT, as before. 

x' 

In short, as these results are all independent of B, the minor 
axis, it follows that the circle and all ellipses on the major axis 
AB can have tangents terminating at the same point T on the 
axis of X, if drawn from the same ordinate, as shown in the 
figure. 

Scholium 2. To find the point in which the tangent to an 
ellipse meets the axis of Y, we make x=0, then the equation 
for the tangent becomes 

B^ 

y 

As this equation is independent of A, it shows that all ellipses 
having the same minor axis, can have tangents terminating in 
the same point on the axis of Y, if drawn from the same abscissa. 

Scholium 3. If from CT we subtract CE, we shall have HT. 
a common suUangerd to a circle, and all ellipses which have 2J 
for a major diameter. That is 

BT=:±^ ~ x'=^ ~^- . 
x' x' 

We can also find RT by the triangle PRT, as we have the 

(B^x'\ 
— I to the radius 1. 
A^y / 

Whence we have the following proportion : 

1 : —^1^=ET : / 
A^y 



RT= 



A'y'^ 



B^x' 

The minus sign indicates that the measure from T is towards 
the left. 



ANALYTICAL GEOMETRY. 



PROPOSITION VII. 

To firid the equation of a normal line to the ellipse. 

Since the normal passes through the point of tangency, its 
equation will be in the form 

y—i/'=a\x—x'). (1) 

Because PJV is at right angles to 
the tangent, 

aa'+l=0. 
But by the last proposition 

a= — , 

Whence a'= — ^, and this value of a! put in (1) give^ 
B'^x 

y — y = — — (x — ^ }> 

for the equation sought. 

Scholium 1 . To find where the normal cuts the axis of X, 
we must make y=0, then we shall have 




-i^-i^y-'"- 



Application. — Meridians on the earth are ellipses ; the semi- 
major axis through the equator is ^=3963. miles, and the semi- 
minor axis from the center to the pole is J5=3949.5. 

A plumb line is everywhere at right angles to the surface, and 
of course its prolongation would be a normal line like PuV. In 
latitude 42°, what is the deviation of a plumb line from the center 
of the earth? Or, how far from the center of the earth would a 
plumb line meet the plane of the equator? Or, what would be 
the value of CiY? 

As this ellipse is very near a circle, we may take CH for the 
cosine of 42°, which must be represented by x'. This being 
assumed, we have 

a;'=2940. (^!^!-) 2940. =23,+ miles CJ}f. Am, 



THE ELLIPSE. |8 

Scholium 2. To find NR, the subnormal^ we simply subtract 
CiVfrom CR, whence 

NR=x'^(^!:^l\x^=^. 
\ A^ / A^ 

We can also find the subnormal from the proportional triangles 
FRT, PNR, thus : 

TR : RP :: RP : RN. 

~:^LI._ : y' :: y' : —NR. Whence iV72=_^. 

PROPOSITIOl^ VIIL 

Z^'w^s drawn from the foci to any point in the ellipse make 
equal angles with the tangent line drawn through the same point. 

Let C be the center of the ellipse, 
PT the tangent line, and PF, PF', 
the two lines drawn to the foci. 

Denote the distance 




CF=JA^—B^ by c, CF' by ~c, the angle FPT by F, and 
the tangents of the angles PTX, PFT, by a and a. 

Now FPT'=zPTX-—PFT. 

By trigonometry, (Eq. 28, p. 143, Robinson's Geometry), we 
have 

Tan. FPT=tsLn.(PTX— PFT). That is, tan. V=-II^. (1) 

l-\-aa' 

Prop. VI, gives us a—— x\ y\ being the co-ordi- 

A^y 

nates of the point P. 

Let X, y, be the co-ordinates of the point F, then from Prop. 

IV, Chap. I, we have 

X — X 

But at the point F, y=0 and x=zc. 
Whence a=^ ^ 



m ANALYTICAL GEOMETRY. 

These values of a and a substituted in ( 1 ) give 

Tan. V^-- ---^^^ 7-^ A^y\x^c)--B^xy' 
A\x'-^) 

Tan F=— ?!^'r:^' =.^1(^—A"J=^, 

(A^—B'')xy'—A''cy' cy'(cx—A'') cy'' 

Observing that A''y'^+B''x"'=A''B\ and A^—B^=c^. 
The equation of the line PF will become the equation of the line 
PF' by simply changing -|-c to — c, for then we shall have the 
co-ordinates of the other focus. 

We now have 

tm.FPT=-^-, 

cy' 

But if c is made — c, then 

tan.i^'P^-—- ^, 
cy 

As these two tangents are numerically the same, differing only 
in signs, they must be equally inclined to the straight line from 
which they are measured, or be supplements of each other. 

Whence FPT+F'PT=1S0, 

But F'PJI+F'PT= 1 80. 

Therefore FPT==F'Pir. Q. E. D. 

Corollary. The normal being perpendicular to the tangent, 
it must bisect the angle made by the two lines drawn from the 
tangent point to the foci. 

Scholium. Any point in the curve may be considered as a 
point in a tangent to the curve at that point. 

It is found by experiment that light, heat, and sound, after 
they approach to, are reflected off", from any reflecting surface at 
equal angles ; that is, any and every single ray makes the angle 
of reflection equal to the angle of incidence. 

Therefore, if a light be placed at one focus of an ellipse, and 
the sides a reflecting surface, the reflections will concentrate at 
the other focus. If the sides of a room be elliptical, and a stove 
is placed at one focus, it will concentrate heat at the other. 



THE ELLIPSE. S6 

Whispering galleries are made on this principle, and all thea- 
ters and large assembly rooms should more or less approximate 
to this figure. The concentration of the rays of heat from one 
of these points to the other, is the reason why they are called 
the foci, or burning points. 

OF THE ELLIPSE REFERRED TO ITS CONJUGATE DIAMETERS. 

Two diameters drawn through the center of an ellipse so as 
to bisect two supplementary chords on the major axis, are said 
to be conjugate. 

Hence, two conjugate diameters intersect one another by an 
angle equal to that of the two supplemental chords, which they 
are supposed to bisect, but by Prop. V, two supplemental chords 
intersect each other by an angle which must conform to the 
equation 

aa = — , 

A'' . 

in which a is the tangent of the angle which one of the supple- 
mental chords makes with the axis of X, and a' is the tangent of 
the angle made by the other chord. 

Now let m be the angle whose tangent is a, and n be the angle 
whose tangent is a, then 

sin. wi J , sin.% 



a= 



— , and 
cos. m cos. n 



Substituting these values in the last equation, and reducing, 
we obtain 

A^ sin. m sin. n-\-B^ cos. m cos. w=0, 

which expresses the relation which must exist between A, B, m, 
and n, to fix the position of any two conjugate diameters in re- 
spect to the major axis, and this equation is called the equation 
of coTidition for conjugate diameters. 

In this equation of condition, m and n are undetermined, show- 
ing that an infinite number of conjugate diameters might be 
drawn, but whenever any value is assigned to one of these angles, 
that value must be put in the equation, and then a deduction 
made for the value of the other angle. 



66 ANALYTICAL GEOMETRY. 

PROPOSITION IX. 

To find the equation of the ellipse referred to its center and con- 
jugate diameters. 

The equation of the ellipse referred to its major and minor 
axes, is 

The formulas for changing rectangular co-ordinates into ob- 
lique, the origin being the same, are (Prop. IX, Chap. I,) 
x-=^x' COS. m-\-y' COS. n. y=ix' sm. m-\-y' cos, n. 

Squaring these, and substituting the values of x^ and y^ in 
the equation of the ellipse above, we have 
{A^sm^ri,-\'B^cos^ri)y'^-\-{A^s\n^m-\-B^cos^m)x^ ) _^2^j 

9.(A^ sin. m sin. n-\'B^ cos. m cos. n)y'x' ) 

But if we now assume the condition that the new axes shall 
be conjugate diameters, then 

A^ sin. m sin. n-\'B^ cos. m cos. w=0, 
which reduces the preceding equation to [(^) 

{A^sm.^n-\-B^'Cos.^n)y'^-\-{AHm.''m-{-B^ cos.^m)x'^=zA^B^ , 
which is the equation required. But it can be simplified as fol- 
lows ! 

The equation refers to the two di- 
ameters B"B' and D"D' as axes. For 
the point B' we must make y'=0, 
then 

^-= ^!^ = 

A'^im.^m-\-B^GOS.^m 
(CB'y=A"-. (P) 
Designating CB' hy A', and CD' by B'. 
For the point D' we must make x'=0. Then 

y'2= -^'^^ =(CI)'y=:B'K (Q) 

A^sin.^n+B^cos.^n ^ ^ 

From (P) we have {A^sm.^m+B^cos.^m)=:^^L 
From (Q) (A''sm.''n+B^cos.^n)=:^l. 




THE ELLIPSE. 



These values put in (F) give 



x'^=:A^Br 



B'^ ' A'^ 

Whence A'^y''+B'''x'^=A'^B'^. 

We may omit the accents to x' and y\ as they are general 
variables, and then we have 

for the equation of the ellipse referred to its center and conju- 
gate diameters. 

Scholium. In this equation if we assign any value to x less 
than A', there will result two values of y, numerically equal, and 
to every assumed value of y less than B\ there will result two 
corresponding values of Xj numerically equal, differing only in 
signs, showing that the curve is symmetrical in respect to its 
conjugate axes, and that each axis bisects all chords which are par- 
allel to the other axis. 

Observation. — As this equation is of the same form as that 
of the general equation referred to rectangular co-ordinates on 
the major and minor axis, we may infer at once thr.t we can find 
equations for ordinates, tangent lines, &c. referred to conjugate 
axes, which will be in the same form as those already found, 
which refer to the rectangular axes. But as a general thing it 
will not do to draw summary conclusions. 

PROPOSITION X. 

As the square of any diameter is to the square of its conjugaiet 
so is the rectangle of any two segments of the diameter to the square 
of the corresponding ordinate; that, is, the ordinate drawn through 
the point of bisection. 

Let CD be represented by A\ and 
CH by B„ CH by x, and GH by y, 
then by the last proposition we have 

A'^y''+B'H''.=A'^B'^. 

Which may be put under the form 

A''y^=B'^{A"'-^x^). 




m ANALYTICAL GEOMETRY. 

Whence A'^ : B'^ : : (A'^'—x^) : y*. 

Or (2Ay : (2By : : (A'+x)(A'—x) : y\ 

Now 2^' and 2-5' represent tlie conjugate diameters D'J), E'E^ 
and since CH represents x, A'-\-x=D'ir, and A' — x=BJ), 
Also y= 6^-5^ Hence the above proportions correspond to 

{D'Dy : (EUy : : D'Ey^HD : (6^^)2. Q. E. D. 

Scholium. As x is no particular distance from C, CF-mAj 
represent x, tlien LF will represent y, and the proportion then 
becomes 

{D'DY : (^'^32 : : D'Fy^FD : (Zi^)2. 

Comparing the two proportions, we perceive that 

D'B'HB : D'F'FD : : "^' : If". 

That is, The rectangle of the abscissas are to one another as tht 
squares of the corresponding ordinates. 

The same property as was demonstrated in respect to rectan- 
gular ordinates in Prop. III. 

In the same manner we may prove that 

Eh'hE' : Ef'fE' : : (hg)^ : (fiy 

PROPOSITION XL 

To find the equation of a tangent line to an ellipse referred to 
its conjugate diameters. 

Conceive a line to cut the curve in two points, whose co- 
ordinates are x', y\ and x'\ y", and ar, y, the co-ordinates of any 
point on the line. 

The equation of a line passing through two points, is of the 
form 

y^y'=a{x-^x'), (1) 

an equation in which a is to be determined when the line 
touches the curve. 

From the equation of the conjugate axes, we have 

A^y'^+B'^x'^::=A'^B'^. 

A'^y^^+B'^x^^^A'^B'^, 



THE ELLIPSE. 69 

Subtracting one of these equations from the other, and ope- 
rating as in Prop. VI, we shall find 

a= — 

This value of a put in (1) will give 

B'^x' ,v 

y—y=^,{x—xy 

A^y 
Reducing, and A'^y'y-\-B'''x'x=A'^B'^, 
which is the equation sought, and it is in the same form as that 
in Prop. VI, agreeably to the observation made at the close of 
Prop. IX. 

PROPosiTioiT xn. 

To transform the equation of the ellipse in reference to conjugate 
diameters to an equivalent equation in reference to its rectangular 



The equation of the ellipse in reference to its conjugate diam- 
eter is 

^'2y3+^'V2 = ^'2^'2. (1) 

And the formulas for passing from oblique to rectangular axes 
are (Prop, X, Chap. I.) 

, xsm.n — yeos.n , ycos.m — xsm.m 
x = , y =- 

sin.(w — m) sm.(n — m) 

These values of x' and y' substituted in ( 1 ) give 

(A'^ COS. ^m+^'2 COS. ^n)y^+(A'^ sin. ^m-\-B'^ sin.^njx' 

— 2(A'^ sin. wi COS. m-\^B'^ sin. n cos. n)xy 

A"" B"" sin.' (nr^m). 

This equation must be true for any point in the curve, x being 
measured on the major axis, and y the corresponding ordinate at 
right angles. 

This being the case, such values of A', B', m, and n, must be 
taken as will reduce the preceding equation to the well known 
form 

A'y'+B^x'=A^B'. 



60 ANALYTICAL GEOMETRY. 

Therefore we must assume 

A'^ cos.^m-\-B''' cos.'n=A^, (1) 

A''' sin. ^m+B'^ sin. ^n=B\ (2) 

A'^ sin. m cos. m-^B'^ sin. n cos. w=0. (3) 

A'^B'^sin.^7i—m)=A^BK (4) 

The values of m and w must be taken so as to respond to the 
following equation, because the rectangular axes are in fact 
conjuffate diameters. 

u4^ sin.msin.w-|-52 cos.mcos. «=0. (5) 

These equations unfold two very interesting properties. 

Scholium 1. By adding (1) and (2) 

Or 4A'''-\-4B'^=4A^-\-4BK 

Thai is, the sum of the squares of any two conjugate diameters is 
equal to the sum of the squares of the axes. 

Scholium 2. Equation (3) or (5) will give us m when n is 
given ; or give us n when m is given. 

Scholium 3. The square root of (4) gives 
^'^'sin. (n—m)=ABy 

which shows the equality of two surfaces, one of which is ob- 
viously the rectangle of the two axes. 
Let us examine the other. 

Let n represent the angle 
^''CB, and m the angle FCB. 
Then the angle NCF will be 
represented by (n — m). 

Since the angle J/lY^is the 
supplement of NCP, the two 
angles have the same sine 
NM=^A'. 
In the right angled triangle NKM, we have 
\ \ A' \\ sm.(n—m) : MK. 




THE ELLIPSE. 61 

MK=zA' sin.(7j— 7»), 
But NC=B'. 

Whence MK- NC=^A' B' ^r[s.,{n — m)= the parallelogram 
NQPM. Four times this parallelogram is the parallelogram 
ML, and four times the parallelogram -D (7^^ which is measured 
by AB, is equal to the parallelogram HF. Hence equation (4) 
reveals this general truth: 

The rectangle which is formed hy drawing tangent lines through 
the vertices of the axes is equivalent to any parallelogram which 
can he found hy drawing tangents through the vertices of conju- 
gate diameters. 



Note.— The student had better test his knowledge in respect to the truths 
embraced in scholiums 1 and 3, by an example : 

Suppose the semi-major axis of an ellipse is 10, and the semi-minor axis 6, 
and the inclination of one of the conjugate diameters to the axis of X is taken 
at 30*^ and designated by m. 

We are required to find il'^ and B'^ , which together should equal 
A^-^B^ , or 136, and the area NCPM, which should equal AB, or 60, if the 
foregoing theory is true. 

Equation (5) will give us the value of n as follows : 

100'|tan.w+36|-^3=0. 



Or tan.w: 



36^3 



100 

Log. 36-f-|-log. 3-.log. 100. Plus 10 added to the index to correspond 
with the tables, gives 9.794863 for the log. tangent of the angle n, which 
gives 31° 56' 42", and the sign being negative, shows that 31° 56' 42" must 
be taken below the axis of X, or we must take the supplement of it, NCB. 
for 7T, whence n=148° 3' 18", and (n— m)=118o 3' 18". 

To find A'^ and B*^, we take the formulas from Proposition IX. 
.,2__ A^B"" __ 100-36 _3600. 



•30-1-52 gos.30 ioo-t+36-f 52 



:69.23. 



^/2_ A^B^ 3600 

-4^8in.^31°66'42"+53cos.a(31°66'42")~27-99+26^" 

66-77 
136.00' 



62 



ANALYTICAL GEOMETRY. 



This agrees with scholium 1, 
As radius 

Is to ^'^(log.69.23) 

So is sine {n—m) 61° 56' 42" 

log. MK— 0.865860 

Log. B'=^ log. (66.77) 0.912290 

^^=60. 



10.000000 
0.920147 
9.945713 



log. 60= 1.778150 




PROPOSITION XIII. 
To find the general polar equation of an ellipse. 

If we designate the co-ordinates of 
the pole P, by a and b, and estimate 
the angles v from the line PX' par- 
allel to the transverse axis, we shall 
have the following formulas : 

x=a-\-r cos.i;. y=5-f-r sin. v. 

These values of x and y substituted in the general equation 

will produce 

^2 sin.^?; T^^^A^bBm.v 1 r+A''h^-\-B^a''=zA^B\ 

JS^cos.^v -^-^B^aco^.v] 
for the general polar equation of the ellipse. 

Scholium 1. When P is at the center, a=0, and 5=0, and 
then the general polar equation reduces to 

^2^ A ^B^ 

A^&m.^'v+B^QO^.^v' 
a result corresponding to equations (P) and ( Q) in Prop. IX. 

Scholium 2. When P is on the curve A^h^+B'^a^zzzA^B^ , 
therefore 



A^^m.^v 
B^cos.^v 



' r=:0. 

-^^B^acos.v] 

This equation will give two values of r, one of them is 0, as 
it should be. The other value will correspond to a chord, 



THE ELLIPSE. 

according to the values assigned to «, h, and v, 
last equation by the equation r=0, and we have 



<5d 

Dividing the 



^^sin.^i 



^^cos.S 



=0. 



-|-2-S-acos.v 

The value of r in this equation is the value of a chord. 
When the chord becomes 0, the value of r in the last equation 
becomes also, and then 

A^ h sin.i;+-B* a cos.v=0. 

Or 



tan.z'= — — - , 
A-'b 



a result corresponding to Prop. VI, as it ought to do, because 
the raditis vector then becomes tangent to the curve. 

Scholium 3. When P is placed at the extremity of the major 
axis on the right, then siu.t'=0, cos.v=l, a=A, and h:=Q. 
These values substituted in the general equation will reduce it 
to B^r^'+^B^Ar^O, 

which gives r=0, and r^ — '2, A, obviously true results. 

When F is placed at either foci, then a=JA^ — B^=Cy and 
5=0. These values substituted, and we shall have 

It is difficult to deduce the values of r from this equation. 
Therefore we adopt a more simple method. 

Let F be the focus, and FP any 
radius, and put the angle PFD=:^v. 

By Prop. I, of the ellipse, we learn 
that 



FP-. 






(1) 




an equation in which c=:^JA^ — B^ , 

and X any variable distance CD. 

Take the triangle PDF, and by trigonometry we have 

1 \ r \ \ cos.v : c-\'X. 

Whence a?=rcos.w — c. 

This value of x placed in (1), will give 

A I cr COS. t; — c^ 
r—ji-f- 



64 



ANALYTICAL GEOMETRY. 



Whence 
Or 



(A — c eo8.v)r=A^ — c^ 



r=. 



A — c cos.v 

This equation will correspond to all points in the curve by 
giving to cos.t; all possible values from 1 to — 1. Hence, the 
greatest value of r is (A-\-c), and the least value (A — c), ob- 
vious results when the polar point is at F. 

The above equation may be simplified a little by introducing 
the eccentricity. The eccentricity of an ellipse is the distance 
from the center to either focus, when the semi-major axis is 
taken as unity. Designate the eccentricity by e, then 
1 : e=A : c. 



Whence 



c=eA. 



Substituting this value ofc in the preceding equation, we have 

A — eAcos.v 1 — e cos.-y 
This equation is much used in astronomy. 



CHAPTER IV. 
The Parabola. 

Definition. — 1 . A parabola is a plane curve, every point of 
which is equally distant from a fixed point and a given straight 
line. 

2. The given point is called the focm, and the given line is 
called the directrix. 

To describe a parabola. 

Let CD be the given line, and F a gi- 
ven point. Take a square, as DBG, and 
to one side of it, GjB, attach a thread, 
and let the thread be of the same length 
as the side GB of the square. Fasten one 
end of the thread at the point G, the 
other end at F. 




THE PARABOLA. 



65 



Put the other side of the square against the given line, CDy 
and with a pencil, P, in the thread, bring the thread up to the 
side of the square. Slide one side of the square along the line 
CDy and at the same time keep the thread close against the 
other side, permitting the thread to slide round the pencil P. 
As the side of the square, BDy is moved along the line CD, the 
pencil will describe the curve represented as passing through 
the points Fand P, 

GP+PF= the thread. 
OP+PB=z the thread. 

By subtraction PF—PB=0, or PF=PJB. 

This result is true at any and every position of the point P ; 
that is, it is true for every point on the curve corresponding to 
definition 1. Hence, FV=VH. 

If the square be turned over and moved in the opposite di- 
rection, the other part of the parabola, the other side of the line 
FIImsLy be described. 

3. A diameter to a parabola is a straight line drawn through 
any point of the curve perpendicular to the directrix. Thus, the 
line HF\s a diameter ; also, ^6^^ is a diameter ; and all diame- 
ters are parallel to one another. 

4. The point in which the diameter cuts the curve, is called 
the vertex. 

5. The axis of the parabola is the diameter which passes 
through the focus. 

6. The parameter to any diameter is the double ordinate which 
passes through the focus. 

7. The parameter to the principal diameter is sometimes called 
the latus-rectum. 

PROPOSITION I. 

To find the equation of the curve. 

The vertex of the parabola is the zero 
point, or the origin of the co-ordinates. 

The distance of the focus F, in the direc- 
tion perpedicular to BH, is called ^, a 
constant quantity, and when this constant 




66 ANALYTICAL GEOMETRY. 

is large, we have a parabola on a large scale, and when small, we 
have a parabola on a small scale. 

By the definition of the curve, V is midway between F and 
the line BH, and PF=PB, 

Put VD=x and PD=y, and operate on the right angled 
triangle PDF. 

FD=x—\'p, PB=x-\-\p—PF. 

(FDy-\-(PDy=^{PFy. 

That is, {x—^pY+y''={x-\-lpY. 

Whence ^*=2pa;, the equation sought. 

Corollary 1. If we make x=0, we have y=0 at the same 
time, showing that the curve passes through the point F", cor- 
responding to the definition of the curve. 

As y=rh^2pa;, it follows that for every value of x there are 
two values of y, numencally equal, one -|-, the other — , which 
shows that the curve is symmetrical in respect to the axis of X. 

Corollary 2. If we convert the equation (y^=2px) into a 
proportion, we shall have 

X : y : : y : 22>, 

a proportion showing that the parameter of the axis is a third pro- 
portional to any abscissa and its corresponding ordinate. 

PROPOSITION" n. 

The squares of ordinates to the axis are to one another as their 

corresponding abscissas. 

Let X, y, be the co-ordinates of any point P, and x' , y\ the 
co-ordinates of any other point in the curve. 

Then by the equation of the curve we must have 

y^ z=i%px, (1) 

y'^=%px\ (2) 



x\ Q. E. D. 



By 


division 




X 




Wt 


lence 


y' •■ 


y'"^ ; : 


X : 




THE PARABOLA. 67 

PROPOSITIOJf III. 

The lotus-rectum is four times the distance from the focus to 
the vertex. 

Let F VH be a parabola, F the focus, and V 
the principal vertex. FH, at right angles to DF, 
through the point F, is the latus-rectum. 

We are to prove that FB=^4FV. 

In the equation of the curve, (y^=2px) for the 
point F, we must necessarily make x=^p, then 
the equation becomes y=p. That is, 

FF=FI>=2 VF, or FB=4: VF. Q. E. D. 

Corollary. It will be observed that CF and DBsire squares, 
and the line DF or its equal FF is the quantity represented by 
J). It is the same for the same parabola, but different in differ- 
ent parabolas. 

PROPOSITION W. 

To find the equation of a tangent line to the parabola. 

Let the line SPQ cut the parabola 
in two points P and Q. 

Let ir, y, be the general co-ordi- 
nates of any point in the line as S ; 
x\ y' the co-ordinates of the point F; 
and x", y", the co-ordinates of the 
point Q. 

The equation of a straight line which passes through the two 
points, S and P, must be of the form 

y—y'^a{x—x'), (1) 

We require the value of a when SP is tangent to the curve. 

If the same line passes through the two points S and Q, we 
must have 

y—y"=a(x—x"). (2) 

And the same line passing through the two points P and Q will 
require the equation 

y'—y"—a{x—x"). (3) 




68 ANALYTICAL GEOMETRY. 

The two points P and Q being in the curve, also require 
y'^~9,px\ (4) 

And y"^=2px\ (5) 

By subtraction y'^ — y"^=^9,p(x' — x"). 

Or {y'—y")(y'+y")^^p{x'—x") (6) 
Dividing (6) by (3) will give 



2/'+y"= 



^p 



(7) 



Now conceive the line >S'^ to turn on the point P as a center 
until Q flows* into P, then we shall have 

Put this value oiy" in (7), and we find 



y 



(8) 



This value of a put in (1) will reduce that equation to 
yy'—y'^=px--px\ 

But y'^=2px' 

By addition yy'=p>{x-\'X') 

and this is the equation sought, x, y, are the co-ordinates of any 
point in the line, and x\ y\ the co-ordinates of the tangent point 
in the curve. 

Corollary. To find the point in 
which the tangent meets the axis of 
X, we must make 2/=0, this makes 

p{X'\-x')=^0. 

Or x'= — X. 

That is, VD=^ VT, or the sub-tangent is bisected by the vertex. 

Hence, to draw a tangent line from any given point, as P, we 
draw the ordinate FD, then make TV= VD, and from the point 
T draw the line TF, and it will be tangent at P, as required. 

*Flows. These conceptions of motion, to make two quantities equal — 
<:!• one to flow out a little in excess of the other, caused Wewton to adopt the 
came of Fluxions. 




THE PARABOLA. $9 

PROPOSITION V. 
To find the eqitation of a normal line to the paraldla. 
The equation of a straight line passing through the point P is 
y^y'^a{x—x'). (1) (Prop. IV, Eq.(l).) 
Let x^, 2/j, be the general co-ordinates of another line passing 
through the same point, and a' the tangent of its angle, its 
equation will then be 

yy—y'=a:(x^-^x). (2) 

But if these two lines are perpendicular to each other, we 
must have 

aa'=—\. (3) 

The first line being a tangent, makes 

y 

This value substituted in (3) gives 
/ y' 

And this value put in (2j will give 

3/1— /=— -(^1— «') 

for the equation required. 

Corollary 1 . To find the point in 
which the normal meets the axis of 
X, we must make yi=0. Then by 
a little reduction we shall have 

2r—x^ — x'. 

But VC=x^, and VD=x\ Therefore J)C=p, that is. 
The sub-normal is a constant quantity, double the distance between 
the vertex and focus. 

Corollary 2. As TV=VD, and VF^^DC. TF=FC, 

Therefore, if the point F be the center of a circle and radius 
FC, that circle will pass through the point P, because TFC is 
a right angle. Hence the triangle FFTis isosceles. 

Now as V bisects TD, and V£ is parallel to PD, the point B 
bisects TP. Join FB, and that line bisects the base of an 




70 



ANALYTICAL GEOMETRY. 




isosceles triangle, it is therefore perpendicular to that base. 
Hence, we have this general truth. 

If from the focus of a parabola a perpendicular he drawn to any 
tangeni, it will meet the tangent on the axis of Y. 

Also, from the two similar right angled triangles, we have 

TF \ FB '.'. FB \ FV. 'Bf^TF-FV. 

But FV is constant y therefore (BF)^ varies as TF, or as its equal PF. 

Scholium. Conceive a line drawn 
parallel to the axis to meet the curve 
at P\ that line will make an angle 
with the tangent equal to the angle 
FTP, but the angle FTP is equal to 
the angle TPF. Therefore, con- 
ceiving this line to be a line of light, 
its reflection from the point P will take the direction PF, and 
this will be true for every other point in the curve ; hence, if a 
reflecting mirror have a parabolic surface, all the rays of light 
that meet it parallel with the axis, will be reflected to the focus ; 
and for this reason many attempts have been made to form per- 
fect parabolic mirrors for reflecting telescopes 

If a light be placed at the focus of such a mirror, it will re- 
flect all its rays in one direction ; hence, in certain situations, 
parabolic mirrors have been made for lighthouses, for the pur- 
pose of throwing all the light seaward. 

PROPOSITION" VI. 

To find the equation of the parahola referred to a tangent line, 
and the diameter passing through the point of contact, the origin 
being the tangent point. 



Let V be the vertex of the parabola, 
VX the axis, and P the origin of the co- 
ordinates. 

Let VS=x, SM=y. Then 

y^='2,px. (1) 

Put VQ^c, QP=h, PE^x', RM=y\ 
and the angle MRS=m. 




THE PARABOLA. 71 

According to this notation we have 

VS=x=c-\-x'-\-y' cos.m. 
SM=sy=b-\--y' sin.m. 
These values of x and y substituted in (1) will give 
b^ -\-2bi/' sin. m-\-y'^ sin.^m=2pc-|-2p:c'-{-2py'cos.7W. (2) 

Because P is on the curve, b^ =2pc, and because BMis paral- 
lel to the tangent FY, we must have (Prop. IV, Eq. (8). ) 

sin.w p 

cos.m b 
Whence 2by' Bm.m=2py' cos.m. 

This equation subtracted from (2) and b'^=2pc; also sub- 
tracted from (2) will reduce (2) to 

y'^ sin.^?w=2^a:'. 

Or y'2=_J^_^'. 

sin.^m 

If we put — ?- — —2p', we shall have for the equation of 
sm.^m 

the curve referred to the origin P, and the oblique axes FX, FY, 

y'2 =z2pV, 
an equation of the same form as that referred to the vertex and 
rectangular axes. 



Corollary 1. As the equation gives y'=-^j2p'x\ that is, 
for every value of y' two values of x, numerically equal, it fol- 
lows that the axis PX bisects all diameters parallel Xo FY. 

Observe that 2p' may be called the parameter of the axis FX. 

Corollary 2. The squares of the ordinates of any diameter are 
to each other as their corresponding abscissas. 

Let Xf y, and x\ y\ be the co-ordinates of any two points oa 
the curve, then 

y^=z9,p'x, 
y'^:=2p'x\ 

Whence l^=f , or y^ : y'« : : x : x'. Q. E. D. 

y'3 x' 



n 



ANALYTICAL GEOMETRY. 




Scholium. Projectiles, if not disturbed hy the resistance of the 
atmosphere, would describe parabolas. 

Let Q be the origin of a projectile thrown 
in any direction as OP. Undisturbed by 
the atmosphere and by gravity it would con- 
tinue in that line, describing equal spaces in 
equal times. But gravity causes bodies to 
fall in proportion to the squares of the times. 
Hence draw IE, TA, ON, proportional to 
the squares of 01, OT, 00, or in proportion to the squares of 
their equals QE, EA, &c. 
Let OQ=IEz=x. OE=TA=zx\ QE~y. EA=y\ 
Then by the construction 

X : ^'=y2 : y''^. 
But this is the property of the parabola, therefore the curve 
made by a projectile is a parabola. 

PROPOSITION VII. 

The parameter of any diameter is four times the distance from 
the vertex of that diameter to the focus. 

We are to prove that 2p'=4FE. 
Let the angle YPE=m as before. 
Then by (Prop. IV,) 

sin.m p 

C0S.7W b 




(0 



From(l) 
Or 



The co-ordinates of the point F being 
c, b, as in the last proposition, whence 
b^=2pc. (2) 

b'^sm,''m=p^cos.^m. 

:=:p^(l — sm.^m)=p'^—p^8in.-m. 



sm.'m= — ±- = — = — —, — 




But in the last proposition — ^^i_=2p'. Whence sin.^m 

sm.2 

Therefore 
Or 



—I 



THE PARABOLA. 73 

But {€-{-^J=PF. (Prop. I.) Hence 2p\ the parameter 

of the diameter PH, is four times the distance of the origin from 
the focus. 

Scholium. Through the focus F draw a line parallel to the 
tangent FY. Designate FF by x, and FQ by y, then by 
(Prop. VI,) 

But FF^FT. (Prop. V, Cor. 2.) And FR==TF, because 
TFRF is a parallelogram. Whence FR=zFF. But FR^x, 

and Pi^=c-f ^. 



Therefore 4a;=4( c+^Ws/, 



or 0;=::?-. 



2 

This value of a; put in the equation of the curve gives 
y=y, or 2y=2y. 

That is, the quantity 2j?', which has been called the parameter 
of the diameter FRy is equal to the double ordinate passing 
through the focus, corresponding to Def. 6. 

PROPOSITION VIII. 

The area of any segment of a parabola made hy co-ordinates, 
(whether right angled or oblique,) is equal to two-thirds of the 
parallelogram formed by the co-ordinates and their parallels. 

Let FX be any diameter, FT a 
tangent, and QR an ordinate paral- 
lel to it. 

Let the angle TFX, or its equal 
QRX=m. 

Put FR—x, and RQ—y. Then, 
by the equation of the curve we have 

y^=2p'x. (1) 

Now let PS=X'\-hj X is increased by RS a space which we 




74 AN^ALYTICAL GEOMETRY. 

designate by h. In consequence of the increase of a:, y must be 
increased by Tty which we will designate by h. Then iSg'=y-f./t. 
Now the equation of the curve at Q,' is 

{y+^)^=2y(^+^). (2) 

Expanding (2) and subtracting (1), and we hare 
2^2/+F=2j9'^. (3) 

Divide by h, then 

2y+A=2p'g). 

This equation is true whatever may be the values of h and ^, 
and we can take h as smaU as we please. If we take it extremely 
small, we may omit Ic in the first member without any appre- 
ciable error ; then 

Dividing this equation by ( 1 ) and 

y^ kx 
Whence 2lcx=hy. 

Multiply each member by sin.m, then we shall have 
9.x ' k sin.m=A • y sin.m. 

Now observe that k sin.m represents the perpendicular distance 
between the lines T^ and tQf, and that y sin.m represents the 
perpendicular altitude of the parallelogram RSQQ', and as h is 
the base, A'y sin.m is the area of that parallelogram, and it is 
equal to two parallelograms whose base is x, and perpendicular 
* sin.m. 

Now the curve space TQR may be considered as made up of 
a great number of parallelograms on the ordinate y, each equal 
to two corresponding parallelograms on the base x. Or any 
parallelogram external to the curve is half of a corresponding 
parallelogram in the curve, therefore the area of the curve is 
double the corresponding external space, or, the area of the seg- 
ment of the curve is two-thirds of the parallelogram formed by 
the co-ordinates of the curve. 



THE PARABOLA. 



76 



The expression for the area of the segment is 
firysin.w. 

Corollary. When the diameter is the 
axis of the parabola, then m=90°, and 
sin.7w=l, and the expression for the area 
becomes f a:y. That is, every segment of a 
parabola at right angles with the axis is two- 
thirds its circumscribing rectangle. 

PROPOSITION IX. 



To find the general polar equation of the parabola. 



Let P be the polar point whose co- 
ordinates are c and b. Put VD=x, 
and I)M=y, then by the equation of 
the curve we have 




■2px. 



(1) 



■m. 




Put PM=R, the angle JlfPZ= 
then we shall have 

Fi)=a:=c+i2 cos.m. 
DM=y^=^h-\-R sin.w. 
These values of x and y substituted in ( 1 ) will give 

(6+i2sin.»i)2=2i?(c+i2cos.m). (2) 

Expanding and reducing, {R being the unknown and variable 
quantity), will give us 

R^ Bm.^m-\-2R{be>m.m — p cos.w)=2^c — h^ 
for the general polar equation of the parabola required. 

CoROLLARr 1. When Pis on the curve, then 5^=2jt?c, and 
the general equation becomes 

R^ sin.^w+2P(5 sin.w— ^cos.m)=0. 
Here one value of R is 0, as it should be, and the other value 
P 2(^cos.m — b&m.m) 



IS 



sm.-'w 



n ANALYTICAL GEOMETRY. 

When m=90f cos.?w=0, and sin.wi=l. Then this last equa- 
tion becomes 

B= — 2bf a result obviously true. 

Corollary 2. When the pole is at the focus F, then b=0, 
and c= -, and these values reduce the general equation to 

jR2 sin.^w — 2Bpcos.m=p^, 
But sin.^m=l — cos.^wi. 

Whence B^ — E^ cos.^m — 2Epcos.m=p^. 
Or B^ —f' -^^Bp cos.m+i2^ cos. ^m. 

Or B=p-{'B cos.m. 

Whence B= 1 , 

1 — cos.m 

and this is the polar equation when the focus is the pole. 

When m=Of cos.m=l, and then the equation becomes 

B=-^—y or B=?.= infinite, 
1—1 

showing that there is no termination of the curve at the right of 
the focus on the axis. 

When m=90°, cos.wi=0, then B=p, as it ought to be, be- 
cause p is the ordinate passing through the focus. 

When m=180°, cos.w=0, then B=lp, that is, the distance 
from the focus to the vertex is ^p. 

As m can be taken both above and below the axis and the 
cos.m is the same to the same arc above and below, it follows 
that the curve must be symmetrical above and below the axis. 

Scholium. If we take p for the unit of measure, that is, as- 
sume ^=1, then the general polar equation will become 

jB^sin.^m-|-2i2(5sin.m — cos.m)=2c — b^ . 
Now if we suppose m=90°, then sin.m=l, cos.m=:0, and B 
would be represented by the line PM\ and the equation would 
become 

B^+2bB=(2c^b^), 
and this equation is in the common form of a quadratic. 




THE PARABOLA. 77 

Hence, a parabola in which p=l will solve any quadratic 
equation by making c= VJB, JBF=b, then PM' will give one 
value of the unknown quantity. 

To apply this to the solution of equations, we construct a 
parabola as here represented. 

Now suppose we require the value 
of y> by construction, in the following 
equation 

Here 26=2, and 2c— 6^=8. 

Whence 6=1, andc=4.6. 

Lay off c on the axis, and from the 
extremity lay off b at right angles 
above the axis if b is plus, and below 
if minics. 

This being done, we find P is the polar point corresponding to 
this example, and PM'=2 is the plus value of y, and PM= — 4 
is the minus value. 

Had the equation been 

2/2__2y=8, 

then P' would have been the polar point, and P'M'=4 the plus 
value, and P'M= — 2 the minus value. 

For another example let us construct the roots of the follow- 
ing equation: 

y^— 6y=-~7. 

Here 5=— 3, and 2c — b^= — 7. Whence c=l. 

From 1 on the axis take 3 downward, to find the polar point 
P". Now the roots are P"m and P"m\ hoth plus. P"m=1.58, 
and P"m'=4.414. 

Equations having two minus roots will have their polar points 
above the curve. 

When c comes out negative, the ordinates cannot meet the 
curve, showing that the roots would not be real, but imaginary. 

The roots of equations having large numerals cannot be con- 
structed unless the numerals are first reduced. 
6 



78 ANALYTICAL GEOMETRY. 

To reduce the numeral in any equation, as 
y3+72y=146, 
we proceed as follows : 
Put y=nz, then 

2 I 72 146 
n n^ 
Now we can assign any value to n that we please. Suppose 
^=10, then the equation becomes 

g2+7-22=:1.46. 

The roots of this equation can be constructed, and the values 
of 7/ are ten times those of z. 

When the square is completed to a quadratic equation, that 
square may be considered the square of an ordinate to a parabola. 



CHAPTER V. 
The Hyperbola. 



Definitions. — 1. The hyperbola is a plane curve, confined by 
two fixed points called the foci, and the difi*erence of the dis- 
tances of each and every point in the curve from the two fixed 
points, is constantly equal to a ffiven line. 

Remark. — The distance between the foci, is also supposed to 
be known ; and the given line must be less than the distance be- 
tween the foci. 

2. The line joining the foci, and produced, if necessary, is 
called the axis of the hyperbola. 

3. The middle point of the straight line which joins the foci, 
is called the center of the hyperbola. 

4. The eccentricity, is the distance from the center to either 
focus, divided by half the given line. 

5. A diameter is any straight line passing through the center 
and terminated by two opposite hyperbolas. 

6. The extremities of a diameter are called its vertices. 



THE HYPERBOLA. 



79 





According to these definitions F', F, are 
the foci, C the center of the hyperbola, A'^ 
A, the given line, and D'D a diameter. 

The parameter is a double ordinate, pass- 
ing through the focus. The principal par- 
ameter passes through the focus at right 
angles to the axis. 

The definition of this curve suggests the following method of 
describing it mechanically : 

Take a ruler F'H, and fasten 
one end at the point F', on which 
the ruler may turn as a hinge. 
At the other end of the ruler at- 
tach a thread, and let it be less 
than the ruler by the given line 
A' A. Fasten the other end of the thread at F. 

With a pencil, P, press the thread against the ruler and keep 
it at equal tension between the points H and F. Let the ruler 
turn on the point F', keeping the pencil close to the ruler and 
letting the thread slide round the pencil ; the pencil will thus 
describe a curve on the paper. 

If the ruler be changed and made to revolve about the other 
focus as a fixed point, the opposite branch of the curve can be 
described. 

In all positions of P, except when at A or A', PF' and PF 
will be two sides of a triangle, and the diflference of these two 
sides is constantly equal to the difference between the ruler and 
the thread ; but that difference was made equal to the given line 
A' A ; hence, by Def. 1, the curve thus described must be an 
hyperbola. 

PROPOSITION I. 

To find the equation of the curve in relation to the center and axis. 

Let C be the zero point. Put 
CA=A. QF^c. CH=x, and 
PIIz=y. (P being any point in the 
curve). Join PP and Pi?" . Put 
FF=r, and PF'=r'. 




I ANALYTICAL GEOMETRY. 

Now we have two right angled triangles, PHF and PHF\ 
By the definition of the curve we have 

/~-r=:2^. (1) 

The right angled A PSF gives 

r2=(ar— c)2+y^ (2) 

The right angled A PHF' gives 

/2^(^_|.,)2^2/^ (3) 

Subtracting (2) from (3) produces 

Dividing (4) by (1), and we have 

/+r=?^. (5) 

Combining (1) and (5), we find 

r'=^+^, and r=— ^+^. 
A A 

This value of t substituted in (2) gives 



Reducing, we find 

A^^^+(A^-^c^)x'=A^(A^^c^), 
for the equation sought. 

Scholium. As c is greater than A, it follows that (A^ — c*) 
must be negativ^, therefore we may assume this value equal to 
— B^. Then the equation becomes 

This form is preferred to the former one on account of its 
similarity to the equation of the ellipse, — the difi'erence is only in 
the negative value of B^. Because A^—c^ =—B^, A^+B^ —c^. 

Now to show the geometrical 
magnitude af B, take (7 as a cen- 
ter, and CF radius, and describe 
the circle FHF'. From A draw 
Aff&t right angles to CF, Now 
GH—c, OA=A, and if we put 
AII=B, we shall have A'^+B^ 
3SC*, as above. Whence ^iJmust equal B, 




THE HYPERBOLA. 
PROPOSITIO^N^ n. 

To determine the figure of the hyperbola from its equation. 
Resuming the equation 



81 



From which we find 



y- 







If we make x=0, or assign to it any value less than A, the 
corresponding value of y will be imaginary, showing that the 
curve does not exist above or below the line A' A. 

If we make x=zA, then 2/=±0, 
showing two points in the curve, one 
at A, the other at A'. 

If we give to x any value greater 
than Ay we shall have two values of y, 
numerically equal, showing that the 
curve is symmetrical above and below 
the axis A' A produced 

If we now assign the same value to x taken negatively, that is, 
make x^ ( — x), we shall have two other values of y, the same as 
before, corresponding to the left branch of the curve. Therefore, 
the hvo branches of the curve are equal in magnitude, and are in all 
respects symmetrical, except opposite in position. 

Hence, every diameter as DD' is bisected in the center, for any 
other hypothesis woidd be absurd. 

Scholium 1 . If through the center 
C, we draw CD, CD', at right angles 
to A' A, and each equal to B, we can 
have two opposite hyperbolas passing 
through D and D' above and below C, 
as the two others which pass through 
the points A' and A, at the right and 
left of C 

The hyperbolas which pass through 2> and D', are said to be 
conjugate to those which pass through A and A', or the two pair 
are conjugate to each other. 




82 ANALYTICAL GEOMETRY. 

DD' is the conjugate diameter to A' A, and DB' may be less» 
equal, or greaier than A' A, according to the relative values of c 
and A, in Proposition I. 

When B is numerically equal to Ay the equation of the curve 
becomes 

and DD'=zAA. In this case the hyperbola is said to be equi- 
lateral. 

Scholium 2. To find the value of the 'parameter , that is, the 
double ordinate which passes through the focus, we must take 
the equation of the curve 

and make x=.c, then 

A^y^=B^{c''-'A^). 
But we have shown that A'^-^-B^^d', or ^^=c^— ^*. 
Whence A''y''=B\ 

Or Ay=B^ , or 2?/=- 



A 

That is, ^A : 2B : : 2B : 2y, 

Showing that the parameter is a third proportional to the transverse 
and conjugate axes. 

Scholium 3. To find the equation for the conjugate hyper- 
bolas which pass through the points D, D', we take the general 
equation 

A^f~'B^x^=—A^B\ 
and change A into B, and x into y, the equation then becomes 

which is the equation for conjugate hyperbolas. 
PROPOSITION" III. 

To jiiid the equation of the hyperbola when the origin is at the 
vertex of the transverse axis. 

When the origin is at the center, the equation is 
A^y'^—B^x^^-^A'B^. 



THE HYPERBOLA. 



83 



And now if we move the origin to the vertex at the right, we 

must put 

x=A-\-x'. 

Substituting this value of x in the equation of the center, we 
have 

A''y^—B^x'^—2JB^Ax'=0. 

We may now omit the accents, and put the equation under the 
following form, 

y^=^(x'-\-2Ax), 

which is the equation of the hyperbola when the origin is the 
vertex and the co-ordinates rectangular. 

PROPOSITION IV. 

To find the equation of a tangent line to the hyperbola, the origin 
being the center. 

In the first place conceive a line 
cutting the curve in two points, P 
and Q. Let x and y be co-ordinates 
of any point on the line, as S, x' and 
y co-ordinates of the point P on the 
curve, and x" and y" the co-ordinates 
of the point Q on the curve. 

The student can now work through the proposition in precisely 
the same manner as Proposition VI, of the ellipse was worked, 
except using the equation for the hyperbola in place of that of the 
ellipse, and in conclusion we shall find 

A^yy'—B''xx'^—A''B'^, 
for the equation sought. 

Corollary. To find the point in 
which a tangent line cuts the axis of 
X, we must make y=0, in the equation 
for the tangent; then 

x=-^=(]T. 
x' 

If we subtract this from C/>, (a;') we 





shall have 



TD=^x'- 



x'^—A' 



84 ANALYTICAL GEOMETRY. 

PROPOSITIO]!^ V. 

To find the equation of a normal to ike hyperbola. 

Let a be the trigonometrical tangent of tlie line TP, (see last 
figure, ) and a' the trigonometrical tangent to the line PN. Then 
if PNis a normal, it must be at right angles to PT, and hence 
we must have 

aa'+l=0. (1) 

Let x' and y' be the co-ordinates of the point P on the curve, 
jind X, y, the general co-ordinates of any point on the line PN, 
then we must have 

y—y'=za'{x—x'). (2) 

In working the last proposition, for the tangent line PT we 
should have found 

A^y' 
This value of a put in (1) will show us that 

a = — — ^. 
And this value of a' put in (2) will give us 

for the equation of the normal required. 

Corollary. To find the point in which the normal cuts the 

axis of X, we must make y=0. 
This reduces the equation to 

Whence x= ^d!±:?!^a;'= CK 

If we subtract CD, («'), from CiV, we shall have DN, the 
stib-normal. 

That is, /A^-{-B^\ ._^.^ ■^' t^e sub-n&rmal 

\ A^ / A' 



THE HYPERBOLA. 
PROPOSITION VI. 



85 



A tangent to the hyperbola bisects the angle contained hy lines 
dravmfrom the point of coniact to the foci. 

If we can prove that 
F'P : FF : : F'T : TF, (1) 
it will tlien follow (Theorem 25, 
Book II, Geometry,) that the angle 
F'PT=:^ the angle TPF. 

In Prop. I, of the hyperbola, we 
find that 

ir'P=/=^+^, and FP^T=—A^ 

A^ 




ex 
A' 

_A^ 

X 



(2) 



F'T==F'C-\-CT=c+--, and TF=c- 

X 

We will now assume the proportion 

Multiply the first couplet by A, and the last couplet by x, 
then we shall have 

{A''-\-cx') : {—A''-\-cx) : : {cx-^-A") : xz. 
Observing that the first and third terms are equal, therefore 
xz-=^cx — A"^ . 
A' 



Or 



;— __=ri^. 



Now the first three terms of proportion (2) were taken equal 
to the first three terms of proportion (1), and we have proved 
that the fourth term of (2) must be equal to the fourth term of 
(1), therefore proportion (1) is true, and consequently 
F'PT=TFF. 

Corollary 1. As TT' is a tangent, and P^Y its normal, it 
follows that the angle TFjS^= the angle T'PN, for each is a 
right angle. From these equals take away the equals TPF, 
T'FQ, and the remainder i^PiVmust equal the remainder QPI^- 
That is, the normal line bisects the exterior angle formed by two lines 
drawn from the foci to any point in the curve. 



86 



ANALYTICAL GEOMETRY. 



Corollary 2. The value of GT we have found to be 
and the value of CD is x, and it is obvious that 
\ A \ \ A \ X, 



A^ 



is a true proportion. Therefore (A) is a mean proportional be- 
tween CT and CD. 

A tangent line can never meet the axis in the center, because 
the above proportion must always exist, and to make the first 
term zero in value, we must suppose x to be infinite. Therefore a 
tangent line passing through the center cannot meet the hyperbola 
short of an infinite distance therefrom. 

Such a line is called an asymptote. 



On Conjugate Diameters. 



Definition. — " Two diameters of an hyperbola are said to be 
conjugate to one another when each is parallel to a tangent line drawn 
through the vertex of the other.'" 

According to this definition, OG' and HE' in the adjoining 
figure are conjugate diameters. 

Explanation 1 . — The tangent line 
which passes through the point jfiTis 
parallel to CG. Hence CG makes 
the same angle with the axis as that 
tangent line does. 

If we designate the co-ordinates 
of the point H, in reference to the 
center and axis by x and y' , and a 
the tangent of the angle made by 
the inclination oi CG with the axis, 
then in the investigation (Proposi- 




tion IV,) we find 



a=. 






0) 



THE HYPEKBOLA. 87 

Now if we designate the tangent of the angle which CH 
makes with the axis by a, the equation of CH must be of the 
form 

because the line passes through the center. 

Whence a'=Z_. (2) 

Multiplying ( 1 ) and ( 2) together, and we find 

aa = , 

A' 

to which equation all conjugate diameters must correspond. 

ExPLANATiox 2. — If we designate the angle GOB by n, and 
HCB by m, we shall have 

sin.m , sin.w 



cos. r)i COS. n 

And tan. ?w.tan. w= — -. 

A^ 

PROPOSITION' VII. 

To find the equation of the hyperbola referred to its center and 
conjugate diameters. 

The equation for the center and axis is 

A^y^—B^x^z=—A^B^. 
Now to change rectangular co-ordinates into oblique, the ori- 
gin being the same, we must put 

x=x cos.w-f-y' cos.w. 
And y=a:'sin.??z-|-y' 

These values of x and y substituted in the above general 
equation, will produce 

(sm.''nA^—(iOE.'^nB'')y^-\-{&m,'^7nA^-^GO^.^mB^)x^) _ (1) 
2(sin.msin.n^2 — cos. 7n cos. nB^)x'y' ) — A^B^ 

Because the diameters are conjugate, we must have 

sm.7n sin.w B^ 

cos.m cos.w A'^ 



'' G0S.71.) 

[ Chap. I, Prop. X. 

f sm.w. ) 



88 ANALYTICAL GEOMETRY. 

Whence (sm.msm.nA^ — cos.mcos.?i^*)=0. (k) 

This last equation reduces (1) to [ (2) 

(sin.2%^2__(j(5s2^jy'2_|_(gin.2^^2__cos.2m52)a;'2=— ^2j52^ 

which is the equation of the hyper- 
bola referred to the center and con- 
jugate diameters. 

If we make y'=0, we shall have 



-A^B' 



? 2 \ \ J 



(sin.^m^^ — cos.^mjS^) 
If we make i»'=0, we shall have 



y"= 



(sin.2^^J.2_cos.27^JB2) 




If we put A'^ to represent CH , and regard it as positive, the 
denominator of (3) must be negative, the numerator being nega- 
tive. That is, sm.^7nA^ must be less than cos.^wij?^. 



That is. 
Or 

But 



sin.2m^2^ cos.2«»^2^ 
B 



tan.m<^ 



tan.mtan.?i=- 



B^ 



Whence 



tan. ??.>-— , or sin.-w^^^^^cos.^w. 



Therefore the denominator in (4) is positive, but the numera- 
tor being negative, therefore Cff" must be negative. Put it 
equal to —^'^ 

Now equations (3) and (4) become 

—A^B^ ^,. ^A^B^ 



A'^== 



—B"' 



(sin.^m^^ — COS. ^mB^) 
Or (am,^mA^--cos.^mB^)z 

(Bm.^nA^-^GOS.^nB^): 



( sin . - nA ^ — cos . ^ nB^ ) ' 
-A^B^ 



A'^ 
A^B^ 
B'^ 



THE HYPERBOLA. 89 

Comparing these equations with equation (2) we perceive that 
equation (2) may be written thus : 

Whence A'^y'^—B'^x'' =—A'^ B'^ . 

Omitting the accents of x' and y'y since they are general va- 
riables, we have 

A'^y^—B'^'x^^-^A'^B'^y 
for the equation of the hyperbola referred to its center and con- 
jugate diameters. 

Scholium 1. As this equation is precisely similar to the 
general equation referred to the center and rectangular co- 
ordinates, it follows that all results hitherto determined in 
respect to the center and rectangular co-ordinates will apply to 
conjugate diameters by changing A to A', and B to B'. 

For instance, the equation for a tangent line in respect to the 
center and axes has been found to be 

A^yy'-^B^xx'==^A^B^ . 

Therefore in respect to conjugate diameters it must be 
A'^yy'—B'^xx'=^—A'^B'\ 

and so on, for normals, sub-normals, tangents, and sub-tangents. 

Scholium 2. If we take the equation 

A'^y^—B'^x^=:^A^B'\ 

and resolve it in relation to y, we shall 
find that for every value of x greater 
than A\ we shall find two values of y 
numerically equal, which shows that 
OiY bisects MM and every line drawn 
parallel to MM, or parallel to a tan- 
gent drawn through L, the vertex of 
the diameter A'. 

Observation. — Let the student observe that these several 
geometrical truths were discovered by changing rectangular to 




90 ANALYTICAL GEOMETRY. 

oblique co-ordinates. We will now take the reverse operation, 
in the hope of discovering other geometrical truths. 
Hence the following : 

PROPOSITION VIII. 

To change the equation of the hyperbola in reference to oblig-ue 
co-ordinates, to an equivalent equation in reference to rectangular 
co-ordinates. 

The equation for the hyperbola in respect to oblique co-ordi- 
nates is 

To change oblique to rectangular co-ordinates, the formulas 
are (Chap. I, Prop. X.) 

, xs'm.n — vGos.n , ycos.m — a;sin.m 

sin.(w — m) sin.(w — m) 

Substituting these values of x' and y' in the equation, we shal\ 
have 

A"-^ (y cos.m — x sin.m) ^ B'^ (x sin.n — y cos.w)^ J'2;»'2 

Bm.^(n — in) sm.^(n — m) 

By expanding and reducing, we shall have 
(A"'Gos.^mr-B'^cos,^n)y''-\-{A'^sm.^m-'B'H'm.^n)x^' 
2( — A'^ sin.m cos.m-f-5'^ sin.n Gos.n)xy 
=~A'^B'^ 8in.2 (n—rn), 

which must be a true equation of the hyperbola corresponding 
to the center and rectangular axes. Therefore it must take the 
well known form 

Or in other words, these tivo equations must be, in fact, iden- 
tical, and we must have 

^'2 C0S.2 m— ^'2 C0S.2 7^=^2 ^ | j 

A"" sm.-m—B'^ sm.^n=-^B^ . (2) 

— A'^ am.m COS. ra-\-B'^ sin.n co8.n=0. (3) 
— ^'2^'*sin.2(«^-w)=— ^^J52. (4) 



THE HYPERBOLA. 



By adding (1) and (2), observing that (cos.^m-\-sm.^m)=\, 
we shall have 

Or 4A'^—4B'^=4A^—4B^, 

which equation shows this general geometrical truth: 

That the difference of the squares of any two conjugate diameters 
is equal to the difference of the squares of the axes. 

Hence, there can be no equal conjugate diameters unless 
A=^B, and then every diameter will he equal to its conjugate : that 
is A'=B'. 

J>'2 

Equation (3) corresponds to tan.mtan.»=— -, the equation 

A 

of condition for conjugate axes. 

Equation (4) reduces to 

A'B'sm.{n — m)=AB. 

The first member is the trigonom- 
etrical measure of the parallelogram 
GCHT, and it being equal to ABi 
shows this geometrical truth : 

That the parallelogram formed hy 
drawing tangent lines through the ver- 
tices of conjugate diameters^ is equiva- 
lent to the rectangle formed hy drawing 
tangent lines through the vertices of the axes. 

Remark. — The reader should observe that this proposition is 
similar to (Prop. XI,) of the ellipse, and the general equation 
here found, and the incidental equations (1), (2), (3), and (4), 
might have been directly deduced from the ellipse by changing 
B into BJ — 1, and B' into B' J — 1. But learners would gene- 
rally demur at results so summarily obtained. 




n ANALYTICAL GEOMETRY. 

On the Asymptotes of the Hyperbola. 

Definition. — If tangents to four conjugate hyperbolas be 
drawn through the vertices of the axes, the diagonals of the 
rectangle so formed and produced indefinitely, are called asymp- 
totes of the hyperbola. 

Let AA', BB\ be the axes of 
four conjugate hyperbolas, and 
through the vertices A, A', B, B\ 
let tangents to the curves be drawn 
forming the rectangle, as seen in 
the figure. The diagonals of this 
rectangle produced, that is, DD' 
and EIJ', are the asymptotes to the 
curve corresponding to the definition. 
If we represent the angle BOX by m, E'CX^iSSS. be m also, 
for these two angles are equal because CB= CB'. 

It is obvious that 

B 




tan.TW: 

A 

sin.^m B'^ 



Whence 

cos.^m A^ 

But cos.*m=l — sin.-7». Therefore 

sin.^m ^2 



1 — Bin.m A 



J03 A2 

Consequently sm.^m= _, and cos.^m: 



A^+B^ A^+B^ 

which equations furnish the value of the angle which the asymp- 
totes form with the transverse axis. 

PROPOSITION IX. 

To find the general equation of the hyperbola^ referred to il% 
centei' and asymptotes. 

Let GM=x, and PM=^y, Then the equation of the curve 
referred to its center and axes is 




THE HYPERBOLA. 93 

From P draw /'^parallel to CE, 
and P^ parallel to CM. Let CH==x\ 
and HP=y'. 

Now the object of this proposi- 
tion is to find the values of x and y 
in terms of x' and y' , to substitute 
in (1), and then the equation re- 
duced to its most simple form will 
be the equation sought. 

The angle ITCM is designated by m, and because EP is paral- 
lel to CB, and PQ parallel to CM, the angle BPQ=m also. 

Now in the right angled triangle Cllk we have ^A=a;'sin.w, 
and Ch=xcos.m. 

In the right angled triangle PQJIwe have HQ=y'sm.m, and 
PQ:=y'QOS.m. 

Whence Hh — IfQ= Qh=PM=y=x'sm.m — y'sin.m. 

Or y=(x' — ^y')sin.w. (2) 

Ck-\-QP= CM=x=x'cos.in-^-y'co8.m. 
Or x=(x'-\-y')cos.m. (3) 

These values of y and x found in (2) and (3) substituted in 
(1) will give 

A''(x'-^y'yBm.^m--B^x'+y'ycoB.''m=--A^£^. 
Taking the values of sin.^m and cos.^w, previously deter- 
mined, we have 

^ ^ (x'—yy—J%^-(x'+y'Y =-'A^jB^ . 

Dividing by A^B'^, and multiplying by (A^-^B^), will give 
(x'-^yr-(x'+y'r=-(A^+B' ). 
Or ^4xy=::-.(A^^B^). 

Or a;y= — ^^ — , 

4 

which is the equation of the hyperbola referred to its center and 
asymptotes. 

Corollary. As x' and y' are general variables, we may omit 

7 



m 



ANALYTICAL GEOMETRY. 



the accents, and as the second member is a constant quantity, 
we may represent it by M^ . Then 

2/ ' 



xy=M^ , or x=- 



This last equation shows that x increases as y decreases ; that 
is, tlte same curve a])^roaches nearer and nearer the asymptote as the 
distance from the center becomes greater and greater. 

But X can never become infinite until y becomes ; that is, 
the asymptote meets the curve at an infinite distance, corresponding 
to Cor. 2, Prop. VI. 

PROPOSITION X. 

All parallelograms between the asymptotes and the curve are equals 
and each eqwd to |AB. 

Let X and y be the co-ordinates 
corresponding to any point in the 
curve, as P. Then by the equa- 
tion of the curve in relation to the 
center and asymptotes, we have 

xy=M\ (1) 

Also let x' represent Gq, and y' qQ, 
that is, x\ y\ co-ordinates of the 
point Q. Then 

x'y'=M\ (2) 
The angle ^(7i) between the asymptotes we will represent by 
2w. Now multiply equations (1) and (2) by sin.27?z. 
Then we shall have 

rr2/sin.2m=if2 sin.Sm. (3) 

a;y 6in.2m=Jf2 sin.2m. (4) 

The first member of (3) represents the parallelogram CP, and 
the first member of (4) represents the parallelogram CQ ; and 
as each of these parallelograms is equal to the same constant 
quantity, they are equal to each other. 

Now A is another point in the curve, and therefore the paral- 
lelogram AHCD is equal to (M^ sin.2w), and therefore equal to 




THE HYPERBOLA. 95 

CQ, or CP. Hence all parallelograms bounded by the asymp- 
totes and terminating in a point in the curve, are equal to one 
another, and each equal to the parallelogram AHCD, which has 
for one of its diagonals half of the transverse axis of A. 

We have now to show the analytical expression for this paral- 
lelogram. 

The angle HCA^m, ACI)=m, and because Aff is parallel 
to CD, CAH=m. Hence, the triangle CAFI is isosceles, and 
CH=^HA. The angle AHq-=9,m. ISTow by trigonometry 

sin. 2m : A : : sin.m : CJI. 
But sin. 2m=2 sin.m cos.m. Whence 

2sin.mcos.m : A : : sin.m : 6W. 
A 



CJI= 



:cos.m 



Multiply each member of this equation by CA=A and sin.m, 
then ■ 

4 / /^Tr\ • -^ Sm. ?i .XX. , 

A.(CJI)sm.m= = tan.m. 

2 cos.m 2 

The first member of this equation represents the area of the 

p 
parallelogram CHAD, and the tan.m= — Hence the parallel- 

A"" B_ 
ogram is equal ~^*~^j — \^B, which is the value also of all the 

other parallelograms, as CQ, CP, &c. 

Scholium. When the asymptotes and any point in the curve 
are given, other points may be determined by the equation. 

For instance, let the asymptotes be given in position, and Q a 
given point in the curve whose co-ordinates are x' and y' , then 

xy'=zM^ . 

Let Cp, any assumed distance, be represented by h, and pP 
by y, then 

hy=M^=x'y\ 
Or y=f>_'. 




m ANALYTICAL GEOMETRY. 

That is, let the numerical value of pP be equal to — -, then 

b 

P will be a point in the curve — and thus any other point may 

be found when the distance along the asymptote is given. 

PROPOSITION XI. 

To find the equation of a tangent line to the hyperhola referred 
10 its center and asymptotes. 

Let X, y, be the general co-ordi- 
nates of a straight line passing 
through the two points P and Q. 

Then the equation of the line must 
be of the form 

y=ax-\-l. (1) 

The same line passing through the 
point P, whose co-ordinates are x\ 
y\ must be 

y'=zax'-\-b. (2) 
And the same line passing through the point Q, whose co- 
ordinates are x", y", must be 

y"=ax"+b. (3) 

Subtracting (2) from (1), and 

y-~2/'=a(ar— «'). (4) 

Subtracting (3) from (2), and 

y'—f=.a{x'-^x"), (5) 

Now the object is to find the value of a when the line becomes 
a tangent at P. 
From (5) we have 

x'^x" 
Which value of a substituted in (4) gives 

y-y'=.i;::f.Xx-x'y (6) 

X X 

But because P and Q are points in the curve, we have 



THE HYPERBOLA. 97 

From each member of this last equation subtract x'y"^ then 
xy' — x'y"=^x''y'' — x'y". 
Or x\y'—y")=-y"{x'^x"). 

Whence V^'i^—^. 



X — X 



This value of the tangent angle put in (6) gives 



y—y 



_ y 



j(x—x'). 



(7) 



Now if we suppose the line to revolve on the point P as a 
center until Q coincides with P, then the line will be a tangent, 
and x'=^x'\ and y'^y", and (7), will become 

y—y ——— {^—-x ), 

X 

which is the equation sought. 

Corollary. To find the point in which 
the tangent line meets the axis of X, we 
must make y=0 ; then 

That is, Ct is double CR, and as RP 
and Cr are parallel, tP^PT. 

A tangent line included between the asymp- 
totes is bisected by the point of tangency. 

Scholium. From any point, as D, draw DG parallel to Tt, 
and from C draw CP, and produce it to S. 

By Scholium 2, to Prop. VII, we learn that CP produced will 
bisect all lines parallel to tT and within the curve ; hence gd is 
bisected in S. 

But as CP bisects tT, it bisects all lines parallel to tT within 
the asymptotes, and i>6^ is also bisected in S ; hence dD= Gg. 

In the same manner we might prove dh=lcVy because hk is 
parallel to some tangent which might be drawn to the curve, the 
same as i> 6^ is parallel to the particular tangent tT. 

Hence, If any line he drawn cutting the hyperbola, the parts be- 
tween the asymptotes and the curve are equal. 




9:8 



ANALYTICAL GEOMETRY. 



This property enables us to describe the hyperbola by points, 
when the asymptotes and one point in the curve are given. 

Through the given point d, draw any line, as DG, and from 
G set off Dg=dD, and then g will be a point in the curve. 
Draw any other line, as hJc, and set off lcv=zdh, then v is another 
point in the curve. And thus we might find other points be- 
tween V and g, or on either side of v and g. 



PROPOSITION XII. 

To find the polar equation of the hyperbola^ the pole heing at 
either focus. 

Take any point P in the hyper- 
bola, and let its distance from the 
nearest focus be represented by f, 
and its distance from the other 
focus be represented by r'. 

Put CH=x, CF=c, and 04=^. 
Then by Prop. I, we have 




A I ex 



(1) 



r'=A-\-. 



ex 



(2) 



Now the problem requires us to remove the symbol x, and 
replace its value by some quantity expressing the value of the 
sine or cosine which r and ?'' make with the transverse axis. 

1st. In the right angled triangle PFH, if we designate the 
angle PFHhj v, we shall have 

1 : r : : cos.?; : PH'=r cos.v. 
CH= QF-\-FH. That is, x=zc-\-r cos. v. 
The value of x put in (1), gives 

^ A 

e—A^ 



Whence 



■ccos.-y 



(3) 



2d. In the right angled triangle F'PH^ if we designate the 
angle PP'Rhj v\ we shall have 

1 : / : : cos.?;' : F'R=r' cos.v'. 



THE HYPERBOLA. .99 

But F'H=F'C-\'CH. That is, rQOB.v'=c-\-x. 
Or x=^rcos„v' — c, and this value of .r put in (2) gives 
/_^ I cr^cos.v'— -c2 
- "^ A 

A 2 f.2 

Whence r'=_± . (4) 

A — CC0S.2;' 

Equations (3) and (4) are the polar equations required. 
Let us examine (3). Suppose v=0, then cos.i;=l, and 

r=tzZ:^=—A—c 
A — c 

But a radius vector can never be a minus quantity, therefore 
there is no portion of the curve in the direction of the axis to 
the right of F. 

To find the length of r, when it first strikes the curve, we 
find the value of the denominator when its value first becomes 
positive, which must be when A becomes equal or greater than 
ccos.v ; that is, when the denominator is 0, the value of r will 
be real and infinite. 

If A — ccos.'y=0, 

Then gos.v= — 

c 

This equation shows that when r first meets the curve, it is 

parallel to the asymptote, and infinite. 

When ?;=90°, cos.v=0, and then r is perpendicular at the 

point F, and equal to , or , half the parameter of the 

A A 

curve, as it ought to be. 

When r=180°, then cos.'y= — 1, and — c cos. i;=c ; then 

f.2 J^2 

r= =c — A^=FAy a result obviously true. 

c+A ^ 

Now let us examine equation (4). If we make v=0, then 
, A^—c^ 



-A-\-c=F'A, as it ought to be. 



100 ANALYTICAL GEOMETRY. 

To find when /will have the greatest possible value, we must 
put 

A — ccos.v'=0. 

A 

Whence cos.v'= — 

c 

Showing, that v' is then of such a value as to make r' parallel 
to the asymptote J and infinite in length. If we increase the value 
of v' from this point, the denominator will become positive, 
while the numerator is negative, which shows that then / will 
become negative, indicating that it will not meet the curve. 



General Remarks. 

When the origin of co-ordinates is at the circumference of a 
circle, its equation is 

When the origin of a parabola is at its vertex, its equation is 
y^=^^px. 

When the origin of co-ordinates of the ellipse is at the vertex 
of the major axes, the equation of the curve is 

y^=z?l^{9.Ax—'X^), 

When the origin of co-ordinates is on the vertex of the hy- 
perbola, the equation for that curve is 

But all of these are comprised in the general equation 

y^=i^px-\-qx^ , 

In the circle and the ellipse q is negative ; in the hyperbola 
it is positive, and in the parabola it is 0. 



CONSTRUCTION OF CURVES. lOl 

JSECTION II. 

CHAPTER I. 

On tbe geometrical representation of Equations of 
tbe second degree between two variables* 

It has been shown in Chap. I, Sec. 1, that every equation of 
the first degree between two variables may be represented by a 
straight line. 

It has also been shown that the equation of the circle, the 
equation of the ellipse, the equation of the parabola, and the 
equation of the hyperbola, each and all correspond to equations 
of the second degree between two variables ; and hence, we 
might naturally infer that a general equation of the second 
degree must represent one or the other of these curves. 

Within the limits designed for this work, there is not space to 
demonstrate this truth rigorously, but we will illustrate it and 
bring it to the comprehension of the learner, partly by general 
theory, and partly by examples. 

An equation of the second degree, in its most comprehensive 
form, is represented as follows : 

Observe that this equation contains the first and second 
powers of each of the variables, their product, and an absolute 
term, F. 

The co-efiicients A, B, C, &c. may be plus, minus, or zero, 
although they are represented above as plus. 

Resolving this equation in relation to y, we obtain [ ( 



Now whatever value may be assigned to x, the equation wil) 
give the corresponding value of y, and if we assume x to be of 



tm 



ANALYTICAL GEOMETRY. 



such a value as to make the quantity under the radical equal to 
0, we shall have 



—4AC 



,+2BJ) 
—^AE 



+D'- 



AAF 



=0. 



(2) 



And 



^ 2 A 2 A 



(sy 



Equation (3) is the equation of a straight line which can 
easily be constructed, and this line will be the same, whatever 
value may be assigned to x. 

Equation (2) is an equation of the second degree, and there- 
fore it may represent a curve; hence equation (1), which is the 
sum of (3) and (2), will represent a curve branching out of a straight 
line. 

We will illustrate this general equation by the following par- 
ticular example : 

Find or construct the curve represented ly the equation 
y'^—2xy-\-2x^ — 3a;+2=0. 

Here A=\, 5=— 2, (7=2, i)=0, ^=—3, F=2. 
These values substituted in (1) give 

'y=x^\ ^_4a;2 + 1 2a;— 8. 
Or y=x±Lj—x^-\-'3x—2, (1) 

If we put the part under the 
radical equal to zero we shall 
have 

y=x 
And — ip2-j_3a;— 2=0. 

The first of these represents the 
straight line AE, passing through 
the origin A at an angle of 45° 
with the axis of X. 

* When A and B in the original equation, have the same sign, the tangent 
of the angle which the line makes with the axis of X, is minus ; when they 
have unlike signs, that tangent is "plus. 




CONSTRUCTION OF CURVES. 103 

The second part resolved, gives a:=l, or x=2. Tailing x=l, 
equation (1) becomes 

y=l±VO. 
And taking x=2, the same equation becomes 

The first result corresponds to the point JD, the second to the 
point U, and DU is the diameter of the curve. 

To find its conjugate diameter JVJV', we must make x corres- 
pond to the point /, the middle point between A G= 1 and AH=-2. 
Hence we must make x=\^, and substituting this value in equa- 
tion ( 1 ) we have 

2/=li±V— 1+1—2. 

Whence 3/=l|-±i==2 or 1. 

The first result is Zzy=2, the second is IN'=\, and therefore 
N'X^zl, the conjugate diameter sought. 

It is obvious from the figure and the values of lines already 
discovered, that DE=J2. 

If we assign to a; a value greater than 0, and less than 1, the 
value of the expression under the radical ( — x^-\-3x — 2) will be 
negative, and hence its square root is impossible, or imaginary, 
and the corresponding value of y imaginary, showing that the 
ordinate would not in that case meet the curve. Again, if we 
take X greater than 2, we shall find a like result. Hence the 
curve must be between the parallels GG' and IfII\ 

The curve must also be within the parallels LM and L'M'. 
Hence it is an ellipse within the parallelogram LL'M'M, and 
BE 2iVL^ iV^'iVare its conjugate diameters, and their angle of in- 
clination as shown in this example is 45°. 

Now by the well known properties of the ellipse we can find 
the rectangular axes and their inclination from these conjugate 
axes. 

If we simply wish to determine whether the curve or line cuts 
either co-ordinate, we take the equation 

3/2 — 2xi/^2x^—3X'{-2=0, 
and make x=0, then y^= — 2, which makes y imaginary, show 
ing that the curve does not cut the axis of Y. 



J 04 ANALYTICAL GEOMETRY. 

Now if we make y=0 in the equation, we have 

Whence x is imaginary, showing also that the curve does not cut 
the axis of X. 



It appears from the preceding example that the equation 



y=^-iAi-Bx+D)±iA^±^^^ 



^ —4AI! 



x+^" 
^—4AF 



represents a curve on a straight line. 

We now attempt to show the natural and possible variations of 
the curve. 

The part under the radical may be represented as follows : 

JMx^-\-Nx^P. (1) 

The equation of the circle when the origin is at the circum- 
ference, is 

yz=j\Rx-^xK 
Now as ar is a variable quantity it is certainly possible that 



This gives (if+l )a;2-J-(iV— 2i2)a;+P=0, 

a quadratic in which there is nothing impossible or absurd. 

Hence, it is possible that the curve indicated by the quantity tinder 
the radical may be a circle. 

When the origin of the co-ordinates is at the vertex of the 
major axis of an ellipse, the equation for that curve is 



^ ^ A A' 

Now it is possible that 



Mx^+]Srx4-P=^^x —:?-zK 
^ ^ A A'- 

Hence, it is possible that the curve under consideration may be 
an ellipse. 

The equation of the hyperbola, when the origin is at the 
vertex of the curve, is 



^ /2^2 j^u 



CONSTRUCTION OF CURVES. 106 

But it is possible that 

^ ^ A ' ^2 

That is, a is possible that ike curve may be an hyperbola. 
The equation for the parabola is 

y=±ij2px. 
And it is possible that 

Mx''+J^x+F=2px, 
and therefore, it is possible that the curve maybe a parabola. 
It is possible that 

Mx^+Nx+P=:Q. 
Then the curve may still be a circle, provided 
2Rx — x^=zO, also. 

The same consideration may be applied to the ellipse, the 
hyperbola, and the parabola. 
Lastly, in the equation 

Mx^-\-]Srx-\-P=0, 

The values of x may be imaginary, and in that case no lines 
can represent it, and the curve itself will be imaginary. 
In short, the equation 



y=-\A{Bx+D)±\AjMx'+Nx+P, 
represents a curve on the straight line, and that curve may be a 
circle, an ellipse, an hyperbola, or a parabola, or the curve may 
be reduced to a point, and then the equation will represent a 
straight line only, or two parallel lines. 

When Mx^ is affirmative, the curve is an hyperbola; when Mz^ 
is negative, the curve is an ellipse, or a circle; and when that term 
is absent, or zero, the curve is a parabola. 

From the preceding summary we learn that the equation 



y—2x—\d[zJ—x''-\-^X, 

must represent a circle on the straight line, whose equation is 



106 ANALYTICAL GEOMETRY. 

1st. Construct that straight line BC. 

2d. Put ±V— ^^+4a;=0. 

Whence x=0, or 4. 

That is, the curve extends from the 
axis of Fto the distance of plus 4, on the 
axis of X. 

Now take P, the middle point between 
A and 4, and make AP=x, then we shall 
have a;=2, which substituted in the equa- 
tion, gives 

2/=4— 1±2=5 or 1. 

That is, Pm'—b, and Fm=l, showing that mm' =4. But 
nn'=4, therefore the curve is a circle. 




OTHER EXAMPLES. 



1 . Find or construct the curve repres<ented hy the equation 
3/2 _j.2^^_|_3^2_4^_-0. 



Whence ?/= — xztij — '^x{x — 2). 

If we make x=^0, we shall find y=0 at the same time, there- 
fore the curve passes through the origin A. 

In the original equation, if we make y=0 we shall have 

Whence x=^Qy or x=^\^. 

Hence, the point E in the curve is 
11 units distance from A. 
If we put 




^— 2a;(ar— 2)=0, 

We shall have a;=0, and a;=2. 
Take ^6^=2, and through 6^ draw 
GIM' parallel to the axis of Y, the 
point / is in the curve at its extreme 
distance in the direction of the axis of 
AI is one diameter of the curve. 



CONSTRUCTION OF CURVES. 107 

As AG-=^, take AR=^\, for the value of x, and substitute 
that value in the equation, and we shall have 

y=— ld=V2=+0-41 or —2.41. 

From R draw i?iV=0.41 and RN'— — 2.41, and through the 
two points N and N' , draw lines parallel to the diameter AI. 
The curve then must be an ellipse described in the parallelogram 
LL'M'M, and NN', — AI, are its conjugate diameters. 

2. Determine what curve corresponds to the equation 
Resolving in relation to y, we find 

This last equation shows that the curve is a parahola, because 
the quantity under the radical does not contain x^ . 

By making x=zO, we find y=3, showing that the curve meets 
the axis of Y three units above the origin. 

Because the sign under the radical is minus, we must take x 
negative, to render the product positive, and hence we decide 
that the parabola must extend in the direction of x negative, 

3. Determine what curve is represented hy the equation 

y^-{-2xy—2x^—4y~x-\-l0=0. ' (1) 

From whence we deduce 



y=—x+2zhjSx^—3x—6. (2) 

Put the quantity under the radical equal to 0, and the corres- 
ponding values of x are — 1, and -\-2, 

Construct the line BD corres- 
ponding to the equation 

y'=—x+2. 

This line is the diameter of the 
curve. 

Make x=0 in (1), and we shall 
have 

?/2— 4y+10=0. 

In this equation y is imaginary, 
showing that no point of the curve is in the axis of Y. 




108 ANALYTICAL GEOMETRY. 

Because x= — 1 or -\-2, under the radical, if we take AB=:2 
and A£J= — 1, and through B and jE'draw the dotted lines pa- 
rallel to the axis of Y, we shall have the limits of the curve, 
and as BJ) is a diameter of the curve, one point in the curve 
must be at B, and the other at D. Hence the curve has two 
h-anches, and it is an hyperbola. 

We might have determined this before, because the co -efficient 
of the second power of x under the cardinal, is positive. Hence 
we can have two positive values for the quantity under the radi- 
cal, one corresponding to x taken as positive, and another corres- 
ponding to X taken as negative. 

The positive value corresponds to the right branch of the 
curve ; the negative value corresponds to the left branch of the 
curve, and BD is one of its conjugate diameters. 

If we make y=0 in either (1) or (2), the corresponding values 
of X will be -\-2 and — 2^, showing that one branch of the curve 
passes through B, and the other through Gr. 

If we make y=l, the corresponding values of ar will be -[-2.14 
*or — 1.64, defining the points n and n\ and thus other points 
may be defined. 

4. Determine the curve represented by the equation 

y 2 -|-6.ry-{-9a;2— 2y— 6*— 1 5=0. 

Resolving the equation in relation to y, we find 

Whence y+^x — 5=0, or y+3a;+3=0. 

Showing no curve, but two parallel lines at the distance of 8 units 
from each other, measured on the axis of Y. 

5. Determine the curve represented by the equation 

y2__4^y_[_5^2 — 2y+5=0. 

On resolving this equation in relation to y, we shall find that 
(3,— 2a;_-l )2 _|_(a;— 2) 2 =02 . 

This last equation will be recognized as the equation of a 
circle whose radius is zero; that is, the curve is diminished down 
to a point. 



CONSTRUCTION OF CURVES. 109 

6. Determine the curve represented by the equation 

Resolving, we find 



This is the equation of a circle whose radius is J — 3, but 
that is impossible. Such a radius is imaginary, and the curve 
imaginary. 

7. What kind of a curve corresponds to the equation 

Ans. It is a parabola passing through the origin and extending 
in the direction of minus x and minus y. 

8. What kind of a curve corresponds to the equation 

y2_^2xy-{^^—2y—l=^0 ? 

Ans. It is a parabola, cutting the axis of X at the distance of 
— 1 and -|-1, from the origin, and extending in the direction of 
plus X and plus y. 

9. What kind of a curve corresponds to the equation 

Ans. It is a straight line passing through the origin, making 
an angle of 30° with the axis of Y. 

10. What kind of a curve corresponds with the equation 

y^—'2xy+2x^—2y-\-2x=0 ? 

Ans. It is an ellipse limited by parallels to the axis of Indrawn 
through the points — 1, and +1, on the axis of X. 

1 1 . What kind of a curve corresponds with the equation 

y2__2xy+x^-\-2y—2z--\-lr=0 ? 

Ans. It is a straight line cutting the axis of Xal an angle of 
45°, at the point +1 from the origin. 
8 



no ANALYTICAL GEOMETRY. 

12. What kind of a curve corresponds to the equation 

Ans. It is an hyperbola. The axis of Y is midway between 
the two branches. One branch of the curve cuts the axis of X 
at the point — 1 ; the other branch cuts the same axis at the 
point -\-3. 



CHAPTER II. 

On Curves and I^ines corresponding to Equations. 

We have seen that the equation of a straight line is 
y=tx-\-c, 
And that the general equation of a circle is 

The first is a simple, the second a quadratic equation, and if 
we eliminate x in the first equation, and substitute its value fox 
X in the second, we shall have a resulting equation of the second 
degree, which cannot correspond to every point in the straight 
line, nor to every point in the circle, but it will correspond to the 
two points in which the straight line cuts the circle, and to those 
points only. 

And if the straight line should not cut the circle, the values 
of y in the resulting equation must necessarily become imaginary. 
All this has been shown in the application of the polar equation 
of the circle, in Chap. II, Sec. I. 

We are now about to extend this principle another step. The 
equation of the parabola is 

y^=2px, 
an equation of the second degree, and the equation of a circle is 

(xzha)^+(y±:by=:E\ 
also an equation of the second degree. But when two equations 
of the second degree are combined, they will produce an equa- 
tion of the fourth degree. 



m 



CURVES AND LINES. Ill 

But this resulting equation of the fourth degree cannot cor- 
respond to all points in the parabola, nor to all points in the 
circle, but it must correspond equally to both ; hence, it will 
correspond to the points of intersection, and if the two curves 
do not intersect, the combination of their equations will produce 
an equation whose roots are imaginary. 

Let us take the equation y^=9,px, and take p for the unit of 
measure, (that is, the distance from the divertrix to the focus is 

unity,) then x=^—y and this value of x substituted in the 

2 

equation of the circle, will give 

Let the vertex of the parabola 
be the origin of rectangular co- 
ordinates. 

Take AP=x, and let it refer 
to either the parabola or the cir- 
cle, and let PM=y, AF=l, 
AH^a, HC^l, and CM=^R. 

Now in the right angled tri- 
angle CMD, we have 

HP==Cn=x—a, MD=y—h, 
and corresponding to this particular figure, we shall have in lieu 
of the equation above 

Whence y*-]-(4—4a)y''—Shy=4{B''—a''—b''). (F) 

This equation is of the fourth degree, hence it must hnyefour 
roots, and this corresponds with the figure, for the circle cuts 
the parabola in fozir points, M, M', M", and M'". 

The second term of the equation is wanting, that is, the co- 
efficient to 2/3 is 0, and hence it follows from the theory of equa- 
tions, that the sum of the four roots must be zero. 

The sum of two of them, which are above the axis of AX, 
(the two plus roots,) must be equal to the sum of the two minus 
roots corresponding to the points M" and M'". 




m ANALYTICAL GEOMETRY. 

The values of a and b and M may be such as to place the cen- 
ter C in such a position that the circle can cut the parabola in 
only two points, and then the resulting equation will be such as 
to give two real and two imaginary roots. 

Indeed, a circle referring to the same unit of measure and to 
the same co-ordinates, might not cut the parabola at all, and in 
that case the resulting equation would have only imaginary roots. 

In case the circle touches the paraholaj the equation will have two 
equal roots. 

Now it is plain that if we can construct a figure that will truly 
represent any equation in this form, that figure will he a solution to 
the equation. For instance, a figure correctly drawn will show 
the magnitude of PM, one of the roots of the equation. 

We will illustrate by the few following 

EXAMPLES. 

1 . Find the roots of the equation 

2/4— ii.i4y2_6.74y+9.9225=0. 

This equation is the same in form as our theoretical equation 
(F), and therefore we can solve \i geometrically, as follows: 

Draw rectangular co-ordinates, as in the figure, and take 
AFz=\, and construct %h.Q parabola. 

To find the center of the circle, and the radius, we put 
4— 4a=— 11.14, (1) —85=— 6.74, (2) 

And 4(i22—a2— 52)=— 9.9225. (3) 

From(l) a=3.78. From (2) 5=0.88. 

And these values of a and b substituted in (3) give 
jB=3.34, nearly. 

Take from the scale which cor- 
responds to AF=\, AH=a= 
3.78, ^(7=0.88, and from C as 
a center, with a radius equal to 
3.34, describe the circle cutting 
the parabola in the four points 
M, M\ M", and M"\ The dis- 
tance of M from the axis of X is 
+3.5, of M' it is -f 0.7, of M" 




CURVES AND LINES. 113 

it is — 1.5, and of M'" it is — 2.7, and these are the four roots of 
the equation. 

Their sum is 0, as it ought to be, because the equation con- 
tains no third power of ?/. 

2. Find the roots of the equation 

?^4 _^3^3 _j_62/2_|_12y— 72=0. 

This equation contains the third power of y, therefore this 
geometrical solution will not apply until that term is removed. 
But we can remove that term by putting 

y=z—\. 
(See theory of transforming equations in algebra.) 
This value of y substituted in the equation, it becomes 

and this equation is in the proper form. 

Nowput 4— 4a=5f, — 85=9|, and 4(i2^— a2_52)=74if|. 

Whence «=— H^ ^=— fi and ^=4.485. 

These values of a and b designate the point C for the center of 
the circle. From this center, with a radius =4.485, we strike 
the circle cutting the parabola in the two points m and m'. The 
point tn is 2^ units above the axis AX, and the point m' is — 2f 
units from the same line, and these are the two roots of the 
equation. The other two roots are iraaginary, shown by the fact 
that this circle can cut the parabola in two points only. 



If we conceive a circle to pass through the vertex of the pa- 
rabola A, then will 

and this supposition reduces the general equation (F) to 
y*+(4— 4a)3/2— 85y=0. 

Here y—ds=.0 will satisfy the equation, and this is as it should 
be, for the circle actually cuts the parabola on the axis of X. 
Now divide this last equation by this value of y, and we have 

y'+(4— 4a)3/=8^. (G) 



114 ANALYTICAL GEOMETRY. 

Here is an equation of the third degree, referring to a parabola 
and a circle ; the circle cutting the parabola at its vertex for one 
point, and if it cuts the parabola in any other point, that other 
point will designate another root in equation (G). 

It is possible for a circle to touch one side of the parabola 
within, and cut at the vertex A, and at some other point. There- 
fore, it is possible for an equation in the form of (G) to have 
three real roots, and two of them equal. 

Most circles, however, can cut the parabola in A, and in one 
other point, showing one real root and two imaginary roots. 

The theoretical equation (G) can be used to effect a mechani- 
cal solution of all numerical equations of the third degree, in 
that form.* 

We will illustrate this by one or two 

EXAMPLES. 

1. Given y3-|-4y=39, to find the value of y hy construction. 

Put 4— 4a=4, and 85=39. Whence a=0, and b=^, 
These values of a and b designate the point C on the axis of 

i^for the center of the circle, C^=4|, the radius. 
The circle again cuts the parabola in P, and PQ measures 

three units, the only real root of the equation. 

2. Given y^ — 75y=250, to find the values of j by construction. 

When the co -efficients are large, a 
large figure is required ; but to avoid 
this inconvenience, we reduce the co- 
efficients, as shown in Chap. II, Sec. I. 

Thus put y=nz. 

Then the equation becomes 

n^z^-~15nz=9.b0. 

» 75 250 
z^ — ~z= 

* Observe that the second term or y^ in a regular cubic is wanting. Hence 
if any example contains that term it must be removed before a geometrical 
solution can be given. 




CURVES AND LINES. 116 

Now take n—5, then we have 

In this last equation the co-efficients are sufficiently small to 
apply to a construction. 

Put 4__4a=— 3, and 85=2. 

Whence a=lf and b=\. 

These values of a and b designate the point D for the center 
of the circle. DA is the radius. 

The circle cuts the parabola in t, and touches it in T, showing 
that one root of the equation is -|-2, and two others each equal 
to— 1. 

But y=nz. That is, y—b'% or — 5, — 5. 

Or the roots of the original equation are -f-10, — 5, — 5. 

When an equation contains the second power of the unknown 
quantity, it must be removed by transformation before this 
method of solution can be applied. 

3. Given y^ — 48y=128 to find the values ofjby construction. 

Ans. +8, — 4, — 4. 

4. Given y^ — 13y= — 12, to find the values ofjby construction. 

Ans. +1, -|-3, and — 4. 

Conversely we can describe a parabola, and take any point as 
IT, at haphazard, and with HA as radius, describe a circle and 
find the equation to which it belongs. 

This circle cuts the parabola in the points m, n, and o, indi- 
cating an equation whose roots are -{-1, -{-2.4, and — 3.4. 

We may also find the particular equation from the general 
equation 

y^-\'(4—4a)a=:Qb, 

observing the locality of ^, which corresponds to a=3-3 and 
5= — 1, and taking these values of a and b, we have 

y3 — 9.2y= — 8, 
for the equation sought. 



116 ANALYTICAL GEOMETRY. 

Remarks and Observations on the general inter- 
pretation of Eatta'tions. 

In every science, it is important to take an occasional retro- 
spective view of first principles, and none demand this more 
imperatively than geometry, and this conviction will excuse ns 
for reconsidering the following truths so often in substance, (if 
not in words,) called to mind before. 

An equation, geoTYietrically considered, whatever may he its degree, 
is bzU the equation of a point, and can only designate a point. 

Thus, the equation y==:ax-\-b designates a point, which point 
is found by measuring any assumed value which may be given 
to X from the origin of co-ordinates on the axis of X, and from 
that extremity measuring a distance represented by (aX'\-h) on 
a line parallel to the axis of Y. 

The extremity of the last measure is the point designated by the 
equation. If we assume another value for x, and measure again 
in the same way, we shall find the point which now corresponds 
to the value of x. Again, assume another value for x, and find 
the designated point. 

Lastly, if we connect these several points, we shall find them 
all in the same right line, and in this sense the equation of the 
first degree yz=zax-{-b, 

is the general equation of a right line, but the right line is found 
by finding points in the line and connecting them. 

In like manner the equation of the second degree 
y=z^J^Rx—x\ 
only designates a point when we assume any value for x, (not 
inconsistent with the existence of the equation,) and take the 
plus sign. It will also designate another point when we take 
the minus sign. Taking another value of x, and thus finding 
two other points, we shall have four points, — still another value 
of x and we can find two other points, and so on we might find 
any number of points. Lastly, on comparing these points we 
shall find that they are all in the circumference of the same circle, 
and hence we say that the preceding equation is the equation of 
a circle. Yet it can designate only one, or at most two points at 
a time. 



INTERPRETATION OF EQUATIONS. 117 

If we assume diflferent values for y, and find the corresponding 
values of x, the result will be the same circle, because the x and 
y mutually depend upon each other. 

Now let us take the last practical example 
y3_i3y=:— 12, 
and for the sake of perspicuity change y into a?, then we shall have 

Now we can suppose y=0 to be another equation; then will 
y=a;3— 13a:+12, (A) 

be an independent equation between two variables, and of the 
third degree. 

The particular hypothesis that y=0 gives three values to a?, 
(-|-1, -[-3, and — 4,) that is, three points are designated, the first 
Mt the distance of one unit to the right of the axis of Y; the 
second at the distance of three units on the same side of the 
axis of Y', and the third point four units on the opposite side 
of the same axis, and this is all the equation can show until we 
make another hypothesis. 

Again, let us assume y=5, then equation (A) becomes 
5=x^—'\^x-\-\2, or ir3__i3.^_|_7==o, 

and this is in efifect changing the origin five units on the axis of 
Y. A solution of this last equation designates three other 
points from the axis of Y. 

Again, let us assume 2/=10, then equation (A) becomes 
a:3— 13ar+2=^0, 
and a solution of this equation gives three other points. 

And thus we may proceed, assigning different values to y, and 
deducing the corresponding values of x, as appears in the fol- 
lowing table, commencing at the origin of the co-ordinates, 
where y=0, and varying each way. 

y=30.0388 .1?=— 2.0814 

2^=25. x= — 1.1 

y=20. x=—OAO 

y=15. x= — 0.20 

j^=10. x=-{-0A4 

y=5. x=+0.55 



+4.1628 


—2.0814 


+4.03 


—2.91 


+3.80 


—3.41 


+3.70 


—3.50 


+3.52 


—3.66 


+3.3 


—3.85 



118 



ANALYTICAL GEOMETRY. 




Wheny=0. then will x—-{-l. +3. —4. 

y=—5 a;=+1.66 +2.477 —4.14 

2/=— 6.0388 a;=+2.0814 +2-0814 —4.1628 

Taking z/=0, a solution of the 
equation 2/=a;3 — 13a;+12, gives the 
three points a, a, a, on the axis of X. 
Then taking 2/= 5, and a solution 
gives three points b, h, h, on a line 
parallel to the axis of X, and at the 
distance of 5 units above said axis. 

Again, taking 3/= 10, and another 
solution gives the three points c, c, c. 
Now joining the three points ah c, ah c, and a 5 c?, we shall 
have apparently three curves corresponding to the equation of 
the third degree, and thus, if we were hasty in drawing conclu- 
sions, we might assume that every equation of the third degree 
might give three curves, and every equation of the fourth degree 
four curves, &c. &c. hut this is not true. 

If we continue finding points as before, we shall find that the 
three curves {a, h, c,) (a, h, c,) and (a, h, c,) are but diflPerent 
portions of the same curve, and we can now venture to draw this 
general conclusion : 

That an equation involving y, the mdinate to the first "power, and 
the abscissa x to the third power, the axis of X, or lines parallel to 
that axis, may cut the curve in three points. 

From analogy, we also infer that an equation involving x to 
the fourth power, the axis of X, or its parallels, will cut the 
curve in four points ; and an equation involving x to the fifth 
power, that axis or its parallels will cut the curve in five points, 
and so on. 

In the equation under consideration, {y=x^ — 13ar+12), if we 
assume y greater than 30.0388, or less than — 6.0388, we shall 
find that two values of x in each case will become imaginary, 
and on each side of these limits the parallels to X will cut the 
curve only in one point. 

Two points vanish at a time, and this corresponds with the truth 
demonstrated in algebra, *' that imaginary roots enter equations 
in pairs." 



INTERPRETATION OF EQUATIONS. 119 

The points m, m, the turning points in the curve, are called 
maximum points, and can be found only by approximation, using 
the ordinary processes of computation, but the peculiar operation 
of the calculus gives these points at once, and we mention the 
fact here, to show the student the practical importance of that 
higher branch of analytical geometry. 

To find the points in the curve we might have assumed differ- 
ent values of x in succession, and deduced the corresponding 
values of y, but this would have given but one point for each 
assumption; and to define the curve with sufficient accuracy, 
many assumptions must be made with very small variations to x. 
We solved the equations approximately and with great rapidity 
by means of the circle and parabola as previously shown. 

We conclude this subject by the following example : 
Let the equation of a curve be 

from which we are required to give a geometrical delineation of 
the curve. From the equation we have 

X 

The following figure represents the curve which will be recog- 
nized as corresponding to the equation, after a little explanation. 

If x=0, then y becomes infinite, 
and therefore the ordinate at A is an 
asyrnptote to the curve. If AB=b, 
and P be taken between A and JB, 
then PM and Pm will be equal, and 
lie on different sides of the abscissa 
AP, If x=b, then the two values of 
y vanish, because x — 6=0 ; and conse- 
quently, the curve passes through B, 
and has there a duplex point. If AP 

be taken greater than AB, then there will be two values of y, as 
before, having contrary signs, that value which was positive 
before, now becomes negative, and the negative value becomes 
positive. But if AD be taken =a, and P comes to D, then the 
two values of y vanish, because Ja^ — a:^=0. And if^P is 




120 ANALYTICAL GEOMETRY. 

taken greater than AD, then a^ — x^ becomes negative, and the 
value of y impossible; and therefore, the curve does not extend 
beyond D. 

If X now be supposed negative, we shall find 
y=±Ja^ — x^ Xh-^-x^x. 
If X vanish, both these values of y become infinite, and conse- 
quently, the curve has two infinite arcs on each side of the 
asymptote AK. If x increase, it is plain y diminishes, and if x 
becomes =a, y vanishes, and consequently the curve passes 
through E, if AE be taken =^AD, on the opposite side. If a; be 
supposed greater than a, then y becomes impossible; and no part 
of the curve can be found beyond E, This curve is the conchoid 
of the ancients. 



CHAPTER II. 
Straight I^ines in Space. 



Straight lines in one and the same plane are referred to tvx) 
co-ordinates in that plane, — but straight lines in space require 
three co-ordinates, made by the intersection of three planes. 

To take the most simple and practical view of the subject, 
conceive a horizontal plane cut by a meridian plane, and by a per- 
pendicular east and west plane. 

The common point of intersection we shall call the zero point, 
and we might conceive this point to be the center of a sphere, 
and from it will be eight quadrangular spaces corresponding to 
the eight quadrants of a sphere, which extended, would comprise 
all space. 

Horizontally, east and west, we shall call the axis of X. Hori- 
zontally in the direction of the meridian, the axis of Y; and 
perpendicularly in the plane of the meridian, the axis of Z. 
From the zero point horizontally to the right we shall designate 
as plus, to the left minus. 

Along the axis of Y and parallel thereto towards us from the 
zero point, we shall call plus ; from the opposite direction will 
therefore be minus. Perpendicularly from the horizontal plane 
upwards is taken as plus, downward minus. 



STRAIGHT LINES IN SPACE. 121 

The horizontal plane is called the plane of xy, the meridian 
plane is designated as the plane of yz, and the perpendicular 
east and west plane the plane of xz. 

Now let it be observed that x will be "plus or minus, according 
to its direction from the plane of yz, y will be plus or minus, 
according to its direction from the plane xz, and z will be plus or 
minus, according as it is above or below the horizontal xy. 

PROPOSITION I. 
To find the equations of a straight line in space. 

Conceive a straight line passing in any direction through 
space, and conceive a plane coinciding with it, and perpendic- 
ular to the plane xz. The intersection of this plane with the 
plane xz, will form a line on the plane xz, and this is said to be 
the projection of the line on the plane xz, and the equation of 
this projected line will be in the form 

x=az-^a. (Chap. I, Prop. 1.) 
Conceive another plane coinciding with the proposed line, and 
perpendicular to the plane yz, its intersection with the plane yz 
is said to be the projection of the line on the plane yz, and the 
equation of this projected line is in the form 

y=hz-]-p. 

These two equations taken together are said to be equations 
of the line, because the first equation is a general equation for 
all lines that can be drawn in the first projecting plane, and the 
second equation is a general equation for all lines that can be 
drawn in the second projecting plane ; therefore taken together, 
they express the intersection of the two planes, which is the line 
itself. 

For illustration, we give the following examxple: Construct the line 
whose equations are 

y=3s— 2) 




in ANALYTICAL GEOMETRY. 

Make z=0, then x=l, and y= — 2. 
Now take AF=1, and draw Fm 
parallel to the axis of Y, making 
Fm= — 2 ; then 7n is the point in 
the plane xy, through which the 
line must pass. 

Now take z equal to any number 
at pleasure, say 1, then we shall 
have x=3 and2/=l. 
Take AF'=3, F'm'=-\~1, and from the point m' in the plane 

xy erect m'n perpendicular to the plane xy, and make it equal to 

1, because we took z=l, then n is anoiheripomt in the line. 

Join nm and produce it, and it will be the line designated by the 

equations. 

PROPOSITION II. 

To find the equations of a straight line which shall pass through a 
given point. 

Let the co-ordinates of the given point be represented by 
x\ y\ z. 

The equations sought must satisfy the general equations 
x=az-\-a^ ^j^ 

The equations corresponding to the given point are 

x'^=az'-\-a. 2/'=5s'+/?. 

Subtracting (1) from these, we have 

X — x=^a{z' — z)y and y' — y=^h(z' — z), 
the equations required. 

PROPOSITION III. 

To find the equations of a straight line which shall pass through 
two given points. 

Let the co-ordinates of the second point be x", y", z'\ Now 
by the second proposition, the equation of the line which passes 
through the two points, will be 

x" — x'=a(s" — z' ) . 



i 



STRAIGHT LINES IN SPACE. 12! 

x" — x' 



Whence a=- 



And y"—y'=h{^'—z'), h=K—r 

z — z 

Substituting the values of a and I in the resulting equations 
of Prop. II, we have 

.'_.= (^i;) (.'-.). y'-y= U=^ )(.'-.), 
\ z! — z / \ z — z J 

for the equations required. 

PROPOSITION IV. 

To find the condition under which two straight lines intersect in 
space, and the co-ordinates of the point of intersection. 

Let the equation of the lines be 

x=az-\-a, y=zhz-\-^. 

x=a'z+a. y=h'z-\-^'. 

If the two lines intersect, (as they do by hypothesis,) then x 
and y may represent the co-ordinate of the point of intersection; 
therefore by subtraction, we have 

{a—a')z-\-a—a'=0, (^h^h')z-\^^—^'=0. 

Whence, by eliminating 2, we find 
a— a' _ ^—^' 
a — a' h — 6' 
which is the condition under which two lines intersect. 

Now 2=— !ZI1_, and this value of z beins^ substituted in the 
a — a 

first equations, we obtain 

x= — and y=JL 1, 

a — a a — a 

for the value of the co-ordinates of the point of intersection. 

Corollary. If a=a\ the denominators in the second mem- 
ber will become 0, making x and y infinite ; that is, the point 
of intersection is at an infinite distance from the origin, and the 
lines are therefore parallel. 



124 



ANALYTICAL GEOMETRY. 
PROPOSITION v.— PROBLEM. 




To express analytically the distance of a given point from the 
origin. 

Let P be the given point in space; 
it is perpendicular over the point 
Nf which is in the plane xy. 

The angle ^ifJV=90°. Also, 
the angle ANP==^0°. 

Let AM=x, MN=y, NP—z. 

ThenZF^=a;2-|-3/2. 

But AP'' ^AN"" +NF =x''-\ 

Now if we designate AP by r, we shall have 
r^—x'^-^y^+z^, 
for the expression required. 

PROPOSITION \a.— PROBLEM. 

To express analytically the length of a line in space, 

N. B. — The only difficulty a learner can experience will arise 
from a want of the proper perception of the figure projected on 
a plane. Hence, teachers should construct the ^yo^qv pasteboard 
figures, which will give the real and simple representation. 

Let PP'=D be the line in question. 
Let the co-ordinates of the point P 
be X, y, z, and of the point P' be x\ 
y', z\ 

Now MM'=:x'—x=JSfQ, 
QN'=y'-y. 

'NN'^={x'—xY+{y'^yY=PR'' 
P'R=z'—z. 

In the triangle PRP' we have 

Or D^=^{x-xY^{y'-yy-\r{z'—z)\ (1) 

which is the expression required. 




STRAIGHT LINES IN SPACE. 125 

Scholium. If through one extremity of the line, as /*, we 
draw PA to the origin, and from the other extremity P', we 
draw P'S parallel and equal to PA, and join AS, it will be 
parallel to PP', and equal to it, and this virtually reduces this 
proposition to the previous one. This also may be drawn from 
the equation, for if A is one extremity of the line, its co-ordi- 
nates X, y, and z, are each equal to zero, and 

PROPOSITION" VII.— PROBLEM. 
To find the inclination of any lim in space to the three axes. I 

From the origin draw a line par- 
allel to the given line, and the 
inclination of this line to the axes 
will be the same as that of the given 
line. 

The equations for the line pass- 
ing fi'om the origin are 

x=^az, and y==hz, (1) 

Let X represent the inclination 
of this line with the axis of x, Kits inclination with the axis 
of y, and Z the inclination with the axis of z. 

The three points P, JSf, M, are in a plane which is parallel to 
the plane zy, and ^if is a perpendicular between the two planes. 
AMP is a right angled triangle, the right angle at M. 

Let AP=r and AM=^x, Then, by trigonometry, we have 

As r : sin. 90° : : x : cos.X Whence x=rcos.X. 

Also, as r : sin. 90° : : y : cos. Y. "Whence y=rcos. K 

Also, as r : sin. 90° : : z : cob.Z. Whence z=r cos. Z. 

From Prop. Y, we have 

r^^x'^+y^+zK (2) 

Substituting the values of x, y, and 0, as above, we hav« 
r^=r^ coa.^X+r'^coa.'^ F+r^cos.^Z. 

Dividing by r^ will give 

co«.2X+cos.2F-}-co8.2Z=l, (3) 
9 




1^ ANALYTICAL GEOMETRY. 

an equation which is easily called to mind, and one that is useful 
in the higher mathematics. 

If in (2) we substitute the values of a;^ and y^ taken from (1), 
we shall have 

r*=a^22_|^2^3^s^ (4) 

But we have three other values of r^, as follows : 

4.2 

r2= . and 



cos.^X cos.^J^ cos.'^Z 



Whence —I =±^Vl+a2+6^ (6) 

cos.JT 



y — 



cos.F 



=±:zjl+a^+b^. (6) 



And _i--=rb7l+a2+62. (7) 

cos.Z 

In (5) put the value of x drawn from (1), and in (6) the 
value of y from (1), and reduce, and we shall obtain 



cos.X=- 



b " 
COS. Y= — : — 

COS.Z : 



The analytical expressions 
for the inclination of a line 
in space to the three co- 
ordinates. 

The double sign shows two angles supplemental to each other, 
the plus sign corresponds to the acute angle, the minus sign to 
the obtuse angle. 

PROPOSITION VIII. 

To find the inclination of two lines in terms of their separate 
inclinations to the axes. 

Through the origin draw two lines respectively parallel to the 
given lines. An expression for the angle between these two 
lines is the quantity sought. 

Let AP be parallel to one of the given lines, and^§ parallel 
to the other. The angle PAQ is the angle sought. 



STRAIGHT LINES IN SPACE. 



127 



Let the equations of one of these lines be 

And for the other 

x'=^a'z', y'zzzb'z. 

Let AP=zr, AQ=r\ PQ=I), and the angle P^ $= K 
Now in plane trigonometry (Prop. 8, page 150 Geometry,) we 
have 



COS. F= 



2rr' 



(1) 




From Prop. VI, we have 

Expanding, and 

D^==(x'^+y'^+z'^)-^ix^+y^+z'')^ 

2x'x — 2yy — 2z'z. 

But from Prop. Y, we learn that 

And x'^+y'^+z'^=r'K 

Whence 2x'X'\-2y'y-\-2z'z=r^'^r'^ — 1>^ . 

This equation applied to ( 1 ) reduces it to 

rr x'x-\-yy-\-zz 
COS. V— — ' ^^X — 

rr' 
But r and r' may be any values taken at pleasure, their lengths 
will have no effect on the angle F, therefore for convenience, we 
take each of them equal to unity. 

Whence coB.V=x'x-\'y'y-\-z'z. (2) 

But Prop. VII, shows that a:=rcos.J;^ 2/=rcos. 1^ &g. and 
that a?'=/cos.X', y'=/cos. Y\ &c. 

But we have taken »'=1, and/=l, therefore a:=co8.A'i <fec. 
and ic'=cos.X', &c. Therefore 

cos. F=cos.Xcos.X'4-cos.Fcos.y-|-<^os.Zcos.Z'. (3) 
But by Prop. VII, we have 

a a' 



COB,X= 



±<yi+a=*+6^ 



and cos.X'- 



±:Jl+a''+b' 



:,&C. 



Its ANALYTICAL GEOMETRY. 

Substituting these values in (3) we have 



COS. V= 



±(Vi+«=-H''')(Vi+«"+*'^) 

for the expression required. 

The cos. V will be plus or minus, according as we take the 
signs of the radicals in the denominator alike or unlike. The 
plus sign corresponds to an acute angle, the minus sign to its 
supplement. 

Corollary 1. If we make F=90°, then cos. F=0, and the 
equation becomes 

which is the equation of condition to make two lines at right 
angles in space. 

Corollary 2. If we make V=0, the two straight lines will 
become parallel, and the equation will become 

Squaring, clearing of fractions, and reducing, we shall find 

Each term being a square, will be positive, and therefore the 
equation can only be satisfied by making each term separately 
equal to 0. 

Whence a'=a, h'=b, and ab'=a'b. 

The third condition is in consequence of the first two. 



CHAPTER IV. 
On the Equation of a Plane. 

An equation which can represent any point in a line is said to 
be the equation of the line. 

Similarly, an equation which can represent or indicate any 
point in a plane, is, in the language of analytical geometry, the 
e(j^uation of the plane. 




EQUATION OF A PLANE. 129 

The co-ordinates AZ, and AX, designate a plane which we 
call the plane of xz. The equation for any line in this plane, as 
My is in the form z=ax-\-b. 

This equation represents points in 
the line M, but if we assign to h any- 
other value, as h\ we shall have points 
in another line parallel to the line M. 
In short, if in place of the constant 
h we write a numerical variable w, we 
shall have 

z=ax'\-w, ( 1 ) 

an equation which will not only rep- 
resent points in the line M, but points 
also in all lines whicb can be drawn parallel to M in the plane 
xz : that is, it is an equation which can represent any point in 
the plane xz; therefore, it is the equation of that plane. 

Like considerations will give us 

z=hy-\-w , (2) 

for the equation of the plane yz, and 

x=b'y+w\ (3) 

for the equation of the plane xy. 

On inspecting either one of the equations (1), (2), or (3), we 
shall perceive that the equation of a plane must he an equation of 
the first degree between three variables, and if either one of the 
variables becomes constant, or is suppressed, the equation tdll 
become that of a straight line. 

Now if the plane in question, is the plane of xz, or parallel 
thereto, the equation of the plane must contain the two variables 
X, z, and one other ; and similarly for each of the other planes. 
But if we have a plane which neither coincides nor is parallel to 
either plane xz, yz, or xy, but intersects all of them, its equa- 
tion must contain the three variables, x, y, and z, and be of the 
first degree. Therefore it must be of the form 
Ax-\-By+Cz-\-D=^0, 

which is recognized at once by all mathematicians as the most 
general and symmetrical equation of a plane. 



130 



ANALYTICAL GEOMETRY. 



Scholium. This notation being adopted, we can at once draw 
from it the following general truths : 

1st. If we suppose a plane to pass through the origin of the 
co-ordinates, the equation for that point requires that x=0, 
y=0, and z=0, and these values substituted in the equation of 
the plane, will give D=0 also. Therefore, when a plane passes 
through the origin of co-ordinates, the general equation for the 
plane reduces to 

2d. To find the points in which the 
plane cuts the axes, we reason thus : 

The equation of the plane must re- 
spond to each and every point in the 
plane ; the point F, therefore, in which 
the plane cuts the axis of X, must cor- 
respond to 3/=0 and z=0, and these 
values substituted in the equation, re- 
duce it to Ax-\-D=0. 




Or 



:-^=0P, 



For the point Q we must take x=0 and g=0. 
And y- 



-i-""- 



„-£,.0R. 



For the point JR, 

3d. If we suppose the plane to be perpendicular to the plane 
XT, PR' a trace in it may be drawn parallel to OZ, and the 
plane will meet the axis of Z at the distance infinity. That is, 

OR, or its equal ( — _ ) must be infinite, which requires that 

C=0, which reduces the general equation of the plane to 

Ax-\-By-{-D=0, 
which is the equation of the trace or line P Q on the plane XY. 
If the plane were perpendicular to the plane ZX, the plane Q, 

or its equal ( — — j, must be infinite, which requires that B=0, 
and this reduces the general equation to 



EQUATION OF A PLANE. 131 

Ax+Cz-^D=0, 

which is the equation for the trace P-5, and hence we may con- 
clude in general terms, 

Thai when a plane is perpendicular to any one of the co-ordinaie 
planes, its equation is that of its trace on the same plane. 

PROPOSITION IX.— PROBLEM. 

To find the length of a perpendicular drawn from the origin to 
a plane, and to find its inclination with the three rectangular co- 
ordinates. 

Let HFQ be the plane, and from the 
origin draw Op perpendicular to the 
plane ; this line will be at right angles to 
every line drawn in the plane from the 
point p. 

Whence Oi? §=90°, OpB=90°, 
Ojt?P=90°. Let Op=p. 

Designate the angle p OP by X, pOQ 
by Y,andpOEhj Z. 

By the preceding scholium we learn that 

0P=--. 0Q=-^, and 0B=-:^, 
A B C 

Af B, C, and D, being the constants in the equation of a plane. 
Now in the right angled triangle OpP, we have 
OP \ \ : : Op \ cos.Jf: 

That is, ^— \ \ w p : cos.Zi (1) 

XX 

The right angled triangle OpQ, gives 

-^ : 1 ::;> : cos.F. (2) 




The right angled triangle OpR gives 



•^ : I ::p : cos.Z, (3) 



132 ANALYTICAL GEOMETRY. 

Proportion (1) gives us 



i>2 



(2) gives COS.* Y=^^B^, 
and (3) gives cos.* Z= ^ (7* . 



(4) 
(5) 
(6) 



Adding these three equations, and observing that the sum of 
the first members is unity, (Prop. VII, Chap. I, Sec. II,J and 
we have 



Whence 



i>* 



J> 



(7) 



This value of |> placed in (4), (5), and (6), and reduced, will 

give 

A 
cos.X= 



cos. Y- 



JA^+B^+C^ 
B 



C0S.Z=rb- 



JA^+B^+C 
C 



(8) 
(9) 

V^*+J5*+C* ^^^^ 

Expressions (7), (8), (9), and (10), are those sought. 

PROPOSITION X.— PROBLEM. 

To find the analytical expressions for the inclination of a plane 
to the three co-ordinate planes respectively. 

Let Ax-^By-^-Cz-^-D^O be the equa- 
tion of the plane, and let PQ represent 
its trace or line of intersection with the 
co-ordinate plane {xy). 

From the origin draw OS perpen- 
dicular to the trace PQ. ^om pS. OpS 
is a right angled triangle, right angled at 
p, and the angle OSp measures the ir)ol\. 




EQUATION OF A PLANE. 133 

nation of the plane with the horizontal plane (a;y). Our object 
is to find the angle OSp. 

In the right angled triangle FOQ we have found 

A B 



Whence PQ=:^JA^-\-B^. 

Now PS, a segment of the hypotenuse made by the perpen- 
dicular OSy is a third proportional to QP and P 0. Therefore 

^JA'-^B^ : — — : : —— : PS. 
AB^ ^ A A 

D BD 



The other segment QS is a third proportional to PQ and OQ, 
Therefore 

— JA'+B' : — — : : — — : QS. 
AB^ ^ B B ^ 



Or JA^+B^:^A::-^: C^^^^._^^. 

But the perpendicular OS is a mean proportional between these 
two segments. Therefore we have 



JA^+B'' 

Now by simple permutation we may conclude that the perpen- 
dicular from the origin to the trace PB, is 

B 

and that to the trace QP is 

B 



JB^ + C^ 

IVe shall designate the angle which the plane makes with the 
plane of (xy) by (a;y), and the angle it makes with (xz) by (xz), 
and that with (yz) by (ys). 



134 ANALYTICAL GEOMETRY. 

Now the triangle Op S gives 

OS : siii.90° :: Op : sin. OSp. 

That IS — - : 1 : : — : sin. OSp 

Whence sin.^ OSp=sm.^(xy)=: •^^+^^ 

Similarly, sin.^xz)^ A^+C^ 



A^+B^+C^ 



And sin.2(yg)r- B^+C^ 



But by trigonometry we know that cos.2 = l — sin.^. 
Whence cos.2('w)=l — i == &c. 

^ ^^ ^2_j_^2_|_(72 ^2_|.^2_|_(72' 

Whence cos.(^)=---p^^ 

coQ,{xzy 



JA^+B^-^G^ 
cos.(y^)=-y==r^=== 



> Expressions sought 



JA^^B^+C^ J 
Squaring, and adding the last three equations, we find 

cos.^ (a:2/)-|-cos. 2 (ar2)+cos.^ (y2)= 1 . 
That is, the sum of the squares of the cosines of the three angles 
which a plane forms with the three co-ordinate planes, is equal to 
radius square, or unity. 

PROPOSITION^ XI.— PROBLEM. . 

To find the equation of the intersection of two planes. 

Let Ax+By-\-Cz+D=0, (1) 

A'x+B'y+C'z+D'=0, (2) 

be the equations of the two planes. 

If the two planes intersect, the values of x, y, and z, will be 
the same for any point in the line of intersection. Hence, we 
may combine the equations for that line. 



EQUATION OF A PLANE. f^ 

Multiply (1) by C\ and (2) by (7, and subtract the products 
and we shall have 

{AC'—A'C)x-\'(BC'—B'C)y+{DC'--D'C)=^0, 
for the equation of the line of intersection on the plane {xy). 
If we eliminate y in a similar manner, we shall have the equa- 
tion of the line of intersection on the plane (xz)', and eliminating 
X will give us the equation of the line of intersection on the 
plane (yz-) 

PROPOSITION XII.— PROBLEM. 

To find the equation to a perpendicular let fall from a given point 
x\ y', z') upon a given line. 

As the perpendicular is to pass through a given point, its 
equations must be of the form 

x—x'=a{z-^z'), (1) 

y—y=K^—z')* (2) 

in which a and h are to be determined. 
The equation of the plane is 

Ax+By-^- Cz+D=0. 

The line and the plane being perpendicular to each other, by 
hypothesis, their projections on any one of the co-ordinate planes 
will be perpendicular to each other. 

The given plane then projected on the planes [xz) and {yz), 
will give Az-\-Cz-\-'D=^0 for the equation of the trace on {yz). 

C D 
From the former x=. — — z — -^. (3) 

A A ^ ^ 

From the latter yz^—^z——. (4) 

B B 

Now equations (1) and (3) represent lines which are at right 
angles with each other. 

Also (2) and (4) represent lines at right angles with each 
other. 

But when two lines are at right angles, (Prop. V, Sec. I, 
Chap. I, ) and a and a\ their trigonometrical tangents, we must 
have (aa'+ 1=0), 



136 ANALYTICAL GEOMETRY. 

C A 

That is, — a — -|-1=0, or «=-^- 

Like reasoning gives us ^=-pj> and these values put in (1) 
and (2) give 

G • for the equations 

, B , ,x r sought. 

PROPOSITION XIIL—PEOBLEM. 

To find the angle included hy two planes given by their eqtiations. 

Lei Ax+£y-{-Bz-\-J)=0, (1) 

And A'x+B'y+Cz+D'=0, (2) 

be the equations of the planes. 

Conceive lines drawn from the origin perpendicular to each 
of the planes. Then it is obvious that the angle contained be- 
tween these two lines is the supplement of the inclination of the 
planes. But an angle and its supplement have numerically the 
same trigonometrical expression. 

Designate the angle between the two planes by V, then Pro- 
position VIII, in the last chapter gives 

^^^- - ±jlJ^a'+b' Jl+a'^+b'^' ^ ^ 

The equations of the two perpendicular lines from the origin 
must be in the form 

x=aZj y=^bz, 

x=az y=^b'z. 

But because the first line is perpendicular to the first plane, 
we must have 

o=— , and b= — , (Prop. XII.) 

C C 

And the second line perpendicular to the second plane requires 
that 

a'=±, and 5'=:^. 



EQUATION OF A PLANE. 



137 



These values of a, b, and a', b\ substituted in (3) and reduced, 
will give 

COS. F=-h T I- — "^ , 

JA^+B'^ + C^ ^A'^+B'^ + a^ 

for the equation required. 

Corollary. When two planes are at right angles, cos. ^=0, 
which will make 

AA'+BB'+CO'=0 

PROPOSITION XIV.— PROBLEM. 
To find the inclination of a line to a plane. 

Let MN' be the plane given by its 
equation 

Ax+B7/-\- Cz+D—O, 
and let P§ be the line given by its equations 
x-=az-\-a. 
y=zbz+^. 

Take any point F in the given line, and 
let fall FB, the perpendicular, upon the plane ; i?§ is its pro- 
jection on the plane, and FQF is obviously the least angle made 
between the line and the plane, and it is the angle sought. 

Let xz=a'z-\~a\ and y=6'2-J-/5', 

be the equation of the perpendicular FRy and because it is per- 
pendicular to the plane, we must have (by the last proposition) 

Because PQ and FR are two lines in space, if we designate 
the angle included by F, we shall have 

\J^aa'+hb' 




COS. F=: 



(Prop. VIII.) 



But the COS. V is the same as the sin.P§JS, or sin. V, as the 
two angles are complements to each other. 



338 ANALYTICAL GEOMETRY. 

Making this change, and substituting the values of a and b', 
we have 

for the required result. 

CoRROLLARY. When F=0, sin.F=0, and this hypothesis 
gives 

Aa+Bb-{-C=0, 

for the equation of a line when it is parallel to the plane. 

We now conclude this branch of our subject with a few prac- 
tical examples, by which a student can test his knowledge of the 
two preceding chapters. 

EXAMPLES. 

1 . What is the distance between two points in space of which the 
co-ordinates are 

x=3, y=5, z=—-2, «'=— 2, y'=— 1, z'=6. 

Ans. 11.180+. 

2. 0/ which the co-ordinates are 

x—l, y= — 5, z= — 3, x'==4, y'= — 4, z'—l. 

Ans. 5y\) nearly. 

3. The equations of the projections of a straight line on the co- 
ordinate planes (xz), (yz), are 

a;=2s+l, y=Xz^2, 

required the equation of projection on the plane (xy). 

Ans. y=lx — 2|. 

4. The equations of projections of a line on the co-ordinate 
planes (xy) and (yz) are 

9.y=x — 5, and 2y=^ — 4, 

required the projection on the plane (xz.) 

Ans. a:=z+l. 



EQUATION OF A PLANE. 139 

5. Required the equations of the three projections of a straight 
line which passes through two points whose co-ordinates are 

x'=% y'=l, 2'=0, and x"= — 3, y"=0, 2"= — 1. 

What are the projections on the planes {xz) and (y^)? 

Ans. x=52-\-2, y=zg^l. 

And from these equations we find the projection on the plane 

(xy), that is, 5y=«+3. 

(See Prop. Ill, Sec. II, Chap. II.) 

6. Required the angle included between two lines whose equations are 

^Z^lfe} of the 1st, and ;Z!+5.i( of the 2d. 

Ans. F=72° 1' 29". 
(See Eq. (3), Prop. XIII. 

7. Find the angles made by the lines designated hi the preceding 
example, toith the co-ordinate axes. (See Prop. VII.) 

(36° 42' withX, (54° 44' with X, 

a4?w. The 1st line J57° 41' 20" J", 2d line J 125° 16' Y, 

(74° 29' 5" Z, (54° 44' Z. 

8. Having given the equation of two straight lines in space, as 

^"^l^fll of the first, and ^=^+^ I of the second, 
y=22+6[ ' y=z^z-\-fi^ 

to find the value of /3', so that the lines shall actually intersect, and 
to find the co-ordinates of the point of intersection. 

Ans. p^-^^ z=—l. 
(See Prop. IV. Sec. II, Chap. I.) 

9. Given the equation of a plane 

8a;— 3y+2— 4=0, 
to find its intersection with the three axes, and the perpendicular dis- 
tance of the origin to the plane. (Prop. IX.) 

Ans. It cuts the axis of Xat the distance of ^ from the origin; 
the axis of P" at — 1^; and the axis of Z at -]-4. 

The origin is |^ of unity below the plane. 

10. Find the equations for the intersections of the two planes 
(Prop. XI.) 3x—4y+2z—l=:0, 

7x—3y—z-{-5=0. 
-4««. On the plane (ary) 17a;— 10y+9=0. 
On the plane (xz) 12a;— 102+23=0. 



140 ANALYTICAL GEOMETRY. 

1 1 . Find the inclination of these two planes. ( Prop. XIII. ) 

Ans, 41^ 27' 30". 

12. The equations of a line in space are 

a;= — 22+l> and y=3z-\'2. 
Mnd the inclination of this line to the plains represented hy the 
equation (Prop. XIV.) 

8a;—3y+2— 4=0. 

Am. 48^ 13'. 

13. Find tlie angles made hy the plane whose eqzcation is 

8ic— 3y+0— 4=0, 
with the co-ordinate planes. (Prop. X.) 

( 84° 9' 40" with (xy). 

Ans. 4 110° 24' 40" with (xs). 

( 21° 34' with (yg). 

14. Find the equation of a plane heing 

Ax-\-By+Cz+D=0, 
Required the equation of a parallel plane whose perpendicular 
distance is (a) from the given plane. 

Ans. Because the planes are to be parallel, they must have 
the same co-efficients. A, B, and (7. 

In Prop. IX, we learn that the perpendicular distance of the 
origin from the given plane may be represented by 
_ D 

^""^JA^+B^+G^ 
Now, as the planes are to be a distances asunder, the distance 
of the origin from the required plane must be 



D I>+aJA^+B^+C' 

Whence the equation required is 



THE DIFFERENTIAL CALCULUS. 



SECTION I. 

CHAPTER I. 
Definitions and Illustrations. 

The differential calculus may be considered a branch of ana- 
lytical geometry ; more literally, it is a science for computing 
the ratio of small differences. 

Newton and his followers called this qqiqucq fluxions, because 
magnitudes were conceived to flow, thus making an increase or 
decrease, and the amount of increase or decrease was the fluxion 
of the particular magnitude or algebraic quantity. But the 
French, and the moderns who have adopted the French phra- 
seology, call this quantity the differential of the given magnitude. 

In some instances the old English method of illustrating this 
science is most simple, and we shall not entirely disregard it. 
They conceived a line to be extended by the motion of a point at 
its extremity, — a surface to be extended by the motion of a line, 
and a solid to be extended by the motion of a surface. 

To illustrate and explain the object of the calculus, we adduce 
the following questions : 

If a side of a square be increased by a very small quantity, 
what effect will that have on the square itself ? 

If a side of a cube be increased by a very small quantity, how 
much will the cube itself be increased ? 

If the arc of a circle be increased by a very small quantity 
10 141 



142 DIFFERENTIAL CALCULUS. 

what effect will this have on the sine and cosine of that arc ? 
What on its tangent and co-tangent? 

If the base of a known right angled triangle be increased by 
a very small quantity, what effect will that have on the hypote- 
nuse and the acute angles ? 

The sun's motion in longitude along the elliptic has a corres- 
ponding motion in declination ; the ratio of these two motions 
at each and every point is a problem in the differential calculus. 

The calls of astronomy gave birth to this science. 

It is not necessary that every part of a magnitude should 
increase or decrease, and therefore we must have variables and 
constants. For instance, one side of a right angled triangle may 
increase while the other side remains constant, and the hypote- 
nuse increase in consequence of the increase of one side. Or 
the two sides may vary at the same time, the one increase, the 
other decrease, and the hypotenuse remain constant. 

Constant quantities are generally represented by the first let- 
ters of the alphabet, a, b, c, &c. and the variable quantities by 
the final letters as u, v, x, y, kc. 

In any equation, as y=.aXy if an increment is given to ar, y 
will have a corresponding increase, and in that case x is said to 
be the independent variable, and y the dependent variable. That 
is, the variation of y certainly depends on the variation of x. 

Thus far we have merely been giving an idea of what the dif- 
ferential calculus is. 



(Art. 1.) When two or more variables enter into an equation 
one is said to be b, function of the other. 

Thus y=a-\-^Xy y=:3a^x-\'X^, y=ij^ — «^» are three dif- 
ferent equations, and here are three different functions of x. That 
is, y expresses three different functions of x, and might express 
ten thousand other functions as well as these. 

If the foregoing equations be resolved in relation to x, so that 
X stands alone as one member, then we might say that x is a 
function of y, hence x and y are functions of each other. 



DEFINITIONS AND ILLUSTRATIONS. 143 

When we wish to express functions in a brief and comprehen- 
sive manner without designating any particvlar equation, we 
write y=/(a;), which means that y equals some algebraic expres- 
sion in which x, as a variable, is contained, and it is read y 
equals a function of x. The letters /, /, , F, F^, stand in the 
place of the word function. Each indicates a different function 
from the other. 

Thus, f(x.y)=0. F(u.x.y)=:0, &c. 

The first of these is a symbol for an equation containing x 
and y as variables, and every quantity in the first member of 
the equation. The second is also a general symbol for some 
equation containing «, x, and y, as variables, and all in the first 
member of the equation. 

If we had an equation in which u was the first member, and 
equal to some algebraic expression containing x, y, and z, as 
variables, and if it were not necessary to write out the explicit 
functions of x, y, and z, we would indicate it generally. 

Thus u=f(x.y.z). 

Or we may write f(u.x.i/.z)=0. 

If we wished to indicate another equation containing these 

same letters, which might exist in the same time, we would 

write 

F(u.x.y.z)=0» 

If still another, we would write 

Fi(u.x.i/.z)=0, 

Functions are either algebraicy circular, or exponential. 

(Art. 2.) We now commence to form rules to differentiate vXl 
classes of algebraic quantities. 
For- example, we have the equation 
y=zax-\-h. 
What will be the eflfect on y, provided x becomes (or-f-^)? 
Let y' represent what y then becomes, and the equation will 
become 

y'=zax-\-ah-{-b 

But y =aa;+6 

Therefore y' — y=ak 



144 DIFFERENTIAL CALCULUS. 

The first member of this equation is obviously the increment 
of y, whatever be the value of h, and when it is extremely/ small 
in relation to x, (we will not say infinitely small, as that word 
puzzles, and it is unnecessary) then {y' — y) is extremely small in 
relation to y, and in that case wo write dy in place of (y'-^y), 
and dx in the place of h. 

The learner must be particular not to regard d in dy or dx as 
a numeral or a co -efficient. It is a symbol, and is read the dif- 
ferential of y, the differential of ar, &c., as the case may be. 

Thus, in general dy means an extremely small increment or 
decrement of y, and dF would denote that P was a variable and 
dP the amount of the variation. 

In the old English fluxions dx, dy, <fec. are represented by x, 
y, &c. the variable with a point over it. The iQiva^fliixion, differ' 
ential, and derivative, all mean the same thing. There are cases, 
as in derived polynomials in algebra, that it would not do to call 
the derivative a differential, as the increment might be too large. 

For another example. If 

u=a-\-^x-{'Cy'^z, 
and if we suppose x becomes (x-\-h), y becomes (y+^), and z 
becomes (2-\-l), what effect will this have on the value of w? 

In consequence of these increments to the variables x, y, and z, 
the dependent variable u becomes u\ and the equation becomes 

u' = a'{-hx-^hh-\-cy-\-cJk-\-Z']-L 

From this, if we subtract the primitive equation, we shall have 

u' — 2*=57i-f-c^+?. 

If we now suppose h, k, and I, to be extremely minute quan- 
tities, dx must be written for k, dy for k, dz for 7, and du for 
(u'—u,) 

Then du=hdX'\'Cdy-\-dz, 

and this is the differential of the primitive equation. 

Comparing this result with the given equation, we draw the 
following rule for differentiating an equation, or any quantity 
involving only the first power of the variables : 

Rule 1. Change each variable into its differential by simply 
writing dx in place of x, dy in place of j, and so on for any other 



INVESTIGATION OF RULES. 146 

variable, preserving the same constant co-efficients that belong to the 
variable, and drop all constants which stand alone, or such as hav* 
no variable factor. 

We give the following examples under this rule : 

1. Differentiate u=a^-\-3a^x-\-h^^-[-4z. 

Ans. du=Sa^dx-\-b'^dt/+4dM. 



2. Differentiate «f=f+-Z-|-l, 

a 36 



Ans. du=^+^. 
a^3b 



3. Differentiate 3u=Jax-}-4a^y-\'-, 

c 

Ans. Mu=z Ja.dx-\-4a^ dy. 

(Art. 3. ) Let us now investigate and draw out a rule to de- 
termine the differential of the product of two variables. 

Let u=-xy, and now suppose that x becomes (x-\-h), and y 
becomes (y-|-^), and in consequence of these increments, u 
becomes u\ and the equation becomes 

v!z=i (x-\-h ) {y-\-lc) =zxy-\-yh •\'x7c-\-hJc. 

Subtracting the original equation, and we have 
u' — u= •yh-\-'Xh-\-hk. 

If now we suppose h and k to be extremely minute quantities, 
their product hic will be still less, and therefore may be omitted 
when h takes the form dx and k becomes dy. This supposition 
reduces the last equation to 

du=ydx^xdy. 

Comparing this equation with the original one (u=xy)f will 
show the truth of the following rule to differentiate a product : 

Rule 2. Multiply each variable by the differencial of the other 
variable, and add their products. 

We may extend this rule to apply to any number of variables. 
For instance, let 

P=zxyz. 



^ 



146 DIFFERENTIAL CALCULUS. 

Also, let u=-xyy as in the former equation, then 

Taking the diflferential of this last equation by the rule just 
formed, and we have 

dP=:^udZ'\-zdu. 

In this equation, for Uy write its value (a?y), and for du write 
its value (ydx-^xdy). Then 

dP=ysdx-\-xzdy~\'Xyd2. 

This equation furnishes a rule for the differential of the pro- 
duct of three variables, which principle being extended, gives 
the following general rule, which will apply to the product of 
any number of variables : 

- Rule 3. Take the differential of each variable and multiply U 
into the product of all the other variables, and add the several pro- 
ducts together. 

Differentiate the following examples under this rule : 

1 . u=xy-\-xyz. 

Ans. du=ydx-{'Xdy-\-yzdx-{-xzdy-\'Xydz. 

2. u=ty — Sxy-^tx. 

Ans. du=ydt'\-tdy — Sydx — Sxdy-\-tdX'{'Xdt. 

3. u=:vxyz. 

Ans. du==xyzdv-\'Vyzdx-\-vxzdy^vxydz. 

(Art. 4.) If in this last equation we suppose v, y, z, each 
equal to a-, then will w=a;*, and dv, dy, dz, must e^ch equal dx, 
and each one of the four products in the answer will be x^dx. 

Consequently du=4x^dXy and the differential of «* must 
be ^x^dx. 

Let us now test this by another course of operation. 

Let u=x*. 

Now suppose that x becomes (x-\'h)f and in consequence of 
this u becomes u', then 

«'= (x+h) " z=:x*+4x^h+6x^h'' +4xh^+h*. 



MS 



INVESTIGATION OF RULES. 147 

Subtracting the original equation and dividing by A, will give 



Now in case k is taken for an extremely small quantity in rela- 
tion to X, all the terms that contain k in the second member 
are comparatively valueless in respect to (4x^) the first term. 

But in case of an extreme small quantity for h we write {dx) 
for ky and du for (w' — u), therefore 

^=4a;». (1) 

dx 

The same result as before deduced from the consideration of 
products. 

In case h is absolutely zero, dx becomes 0, and du also be- 
comes 0, and equation ( 1 ) becomes 



But there is nothing absurd in this, as we learn by algebra 
that divided by can be any quantity whatever. 

Now (u' — u) represents the increment of the function w; and h 

that of the variable x, and therefore — ZH is the ratio, and this 

h 

ratio diminishes as h diminishes, and comes to a limit when h 

equals 0. 

Therefore _ is the limiting ratio between a function and its 

variable. In this example 4x^ is that ratio, it is also called the 
differential co-effcient. 

For — z=4x^ , or du=4x^dx. 

dx 

Here it is obvious that 4x^ is a co-efficient to dx, the differ- 
ential of the variable. 
For another example, let 

Now as before let x become (x-^-dx), then u becomes «+c?«, 
and the equation becomes 

u+du=(x+dx)''=x'^-{^mx'^^dx+m^!^ x'^^dxy^&c. 



148 DIFFERENTIAL CALCULUS. 

Subtracting the original equation, and dividing by (dx), wo 
have 

^=WM;«-»+m'!!^a;»-Va:+ &c. 
dx ^2 ^ 

Now let us suppose dx=0, that is, pass to the limit, and 

du m„, 

dx 

From this equation we can draw the following general rule to 

find the differential co-efficient of any quantity in the form a;", that 

is, any power of the variable : 

Rule 4. Multiply hy the exponent, and diminish the exponent by 
unity. 

By the first example in this article the learner will perceive 
the truth of this rule when the exponent m is a whole positive 
number, such as x^, x^, &c. &c., and yet not convinced of its 
application when m is fractional or negative. 

But we learn in algebra that (x-^-dx)"^ expands in the same 
form, whatever be the value of m, whole or fractional, positive 
or negative, therefore the rule must be generally applicable, what- 
ever be the value of m. 

For example, what is the differential co-efficient in the following 
equation, in which m is taken both negative and fractional : 

u=x-v^ 
By the rule we have at once 

dx t 

Or the differential of the function u is 

^Ix^'dx. 
t 

Suppose now that we distrust the rule, and require the result 

by a more elementary and obvious process, and if we arrive at 

the same conclusion it would be very unphilosophical to distrust 

it in any future case. 



INVESTIGATION OF RULES. 149 



Resuming M=a: * . This is tlie same as 



w=- 



1 



or ux^=\. 



X t 
Raising each member to the power t, then 

U^X^=z\ (1) 

PutP=w» and Q=x\ then PQ=\. 

Now we can differentiate this equation by (Rule 2,) which 
gives 

PdQ^QdP=Q. (2) 

As t and s are whole positive numbers, 

t— 1 s — 1 

dP=^tu duy and dQ=sx dx. 

Substituting these quantities in (2), and retaining the original 
values of P and Q, equation (2) becomes 

su t a;»-ic?a?+te ^ u ^-^du=0. ( 3) 

Multiply (3) by ux, which gives 

su^x^udx-\-tx'u*xdu=0. (4) 

But equation ( 1 ) gives «* * a; * = 1 . Therefore (4) reduces to 
sudx-\-ixdu=0. (6) 

Or 



du su 



dx tx 

Substituting the value of u in the second member taken from 
the original equation, and we have 

du_ ij'f^ 

dx t 

The same value as given by the rule, and thus the rule could be 
verified in every possible case. 

Rule 4, can be made of very extensive application in the cal- 
culus, as the following examples will show : 

u. =(1+^)*- 



150 DIFFERENTIAL CALCULUS. 

Put (l'^x)=z, then u=z^, an equation to which the rule 
will apply, giving 



dz ^ 2jz 

Because l-\~x=z, dx=:dz. Therefore 



1 



Or 



dx 



2(l+x)^' 
For another example, take 

u=Jl-\-x — x^. 
As before, put l-\-x — x^=z. Then dx — 2xdx=dz, and u=z^, 

B7 the rule, ~=lz ^=^ 



Or du 



dz 2jz 

_ dz _ (l--2x)dx 



From these examples we may draw the following rule to dif- 
ferentiate the square root of any quantity : 

Rule 5. Differentiate the quantity under the radical, and divide 
it hy twice the radical. 

The last equation may be differentiated without substitution, 
thus 



Uz=J\-\-X — x^. 

Square both members, and 

u^ = \-\-x — x^ . 

Now apply the rule to each member, and 
2udu=dx — 2xdx. 
(1 — 2x)dx 



Whence du-. 



^Jl+x—x^ 



INVESTIGATION OF RULES. 161 

The preceding rule maj be made general, as will appear by 
the following example : 

Take the nth. power of each member, then 

Taking the differential of each member, gives us 

%%"" ^du=( h — 2cx)dx. 
Multiply this by the given equation, and we have 

I 

nMu={h — 2cx)dx(a-\-hx — cx^) ^ 

Whence dn~ (i-2cx)dx( a +hx-cx')7 

n[a-\-bx — cx"^) 
From this equation we draw the following general rule to 
differentiate any radical quantity : 

Rule 6. Take the differential of the quantity under the radical ^ 
multiply it hy the radical, and divide the product by n times the 
quantity under the radical, n being the index of the root. 

For an example under this rule we give the following equation: 

By the rule c?^.= (izi^^l^CA+^^Zf!)!. 
^ 6(l+2a;— rc2) 

Results under this rule are always reducible^ as we have the 
same quantity in the numerator and denominator of the second 
member, with different exponents. By subtracting one expo- 
nent from the other, and dividing numerator and denominator 
by 2, we get the following reduced result : 

{\—x)dx 

du^=. L. 

3(l-|-2a;— a:3)6 

(Art. 5.) Sometimes we have u=f[y), and y:=F(x), and 
require the differential co-efficient between the function u and 
the variable x. 



152 DIFFERENTIAL CALCULUS. 

For that purpose we first find ~ from one equation, and then 

J!, from the other, and multiply them together, and we have ~ 
dx dx 

as required. 

For example, suppose «=l+2y+y*, and y=a+a;3, and we 

require the ratio ~y we obtain it thus : 
dx 

From the first ^=2+2y. 

dy 

From the second _J^=3a;* . 

dx 

Multiplying these together and we obtain the final result 

We might have taken the value of y, {a-\-x'^ ) in the second 
equation and substituted it in the first, and then have taken the 
difi*erential, but this would have been more troublesome. 

(Art. 6.) We have but one more rule to advance to enable 
the learner to difi'erentiate all kinds of algebraic guantiiies. That 
is, a rule to differentiate a fraction. 

Let it be required to differentiate the fraction -. Put u=-, 

y y 

Then the value of du will be the differential of the fraction. 

Clear of fractions and uy=x. 

The first member is a product of two variables, therefore dif- 
ferentiate it by Rule 2. 

Whence udy-{-ydu=idx 

Restoring the value of u, this last equation becomes 

— K-l-ydu = dx, 

y 

Or xdy-^-y^du^ydx. 

Whence du=y^J^, 

y^ 



INVESTIGATION OF RULES. 163 

From this equation we can draw the following rule to diflfer- 
entiate a fraction : 

Rule 7. From the differerdial of the numerator multiplied by 
the denominator, subtract the differential of the denominator multiplied 
into the numerator, and divide the difference hy the square of the 
denominator. 



We can obtain the same result, independent of a product, as 
follows : — The fraction to be differentiated is -. Let x become 

y 

z-\-dx and y become y-\-dy. 

Then the fraction becomes .^i-*, and from this subtract the 

original fraction, and we shall have the difference, which is the 
differential in case both dx and dy are extremely small. 

x-\-dx X xy-\'ydx — xy — xdy 

y+dj/ y y^+ydy 

The difference is ^ ^ ^ ^ , whatever he the values of dx and dy; 

y +ydy 

but when dy is extremely small, (not to say infinitely small), then 
y* is not sensibly augmented by the addition of ydy, and there- 
fore the differential of the fraction is ti — ZI — fL, as before found. 

y"" 

The preceding rules and combinations of them will serve to 
differentiate any algebraical expression that can be given. Yet 
there are cases that might appear inapplicable to the rules, at 
first view, and in these operations there is room to exercise 
algebraical tact and skill. 

We give the following examples : 

1, Differentiate the equation 

-As explained in (Art. 5,) put y^ia-^-hx^t then w=y°'. Now 
from this last equation 

^^mf^K (Rule 4.) 

dy^ 



164 DIFFERENTIAL CALCULUS. 

Also ^=35a;^ (Rule 4.) 

By multiplication _ =36??ia;2y"-^». 

Substituting the value of y^~^, and multiplying by dx, and 
du=Sbmx^ (a-\-bx^ )"'"* dx. 

2. Differentiate the equation 

u=x(a^+x^)Ja^^^^. (1) 

This requires the application of Rules 3 and 5, but to show 
independence and tact, put 

y=Ja^—x^. (2) 

Multiply (1) and (2), and we obtain 
uy=a^x — x^. 
Taking the differential of each member, 

udy-\-ydu=a*dx — 5x*dz, (3) 

From (2) we find 

The product of (4) and (1) is 

udy='-{a^x^+x* )dx, (&) 

Subtracting (6) from (3), and we have 

ydu=(a^-\'a^x^—4x^ )dx. 

Finally, <,„=Li^_=_^^. 

Some expressions may be reduced or changed in form to ad- 
vantage before attempting to take the differential. The following 
is an example of the kind : 



J\-\-x-X-J\ — X 

3. Given u= — ==. to find ike differential of u. 

^l+a; — ^1 — X 

Reduce the second member by multiplying numerator and 
denominator of the fraction by the numerator, then 



INVESTIGATION OF RULES. 165 



-x^dx 



Whence du=. ^-==^-dx{X+.J^-x^) ^^^^^ ^^ 
Dividing by rfar, and changing signs, we have 
dx JU^'^ x^ 



J{.J\—x^Ji^\^ x^_ \+J\—x^ 
Whence du=- — • _ 1 dx. 



x^J\—x' 

We add the following unwrought examples to exercise the 
powers of the learner : 

4. «= aM= . 

x^ x^ 

5. tt=-^ du=^:z^. 

{xf ^x^ 

6. u=^x^y^ du=:2x'ydy-{-2y^xdx. 

7. u=j2ax+x^,.JRu\e 6, reduced. )..^m= (^jl^^ . 



8. M=. 



dx 



9. u= ^ du=. "^^"^ 



sJl-'X' (1— a;2)t 

10. tt= ^ du=. ^"^ 



1+a:* , __ 4xdx 



11. «=__L__ c?t<= 



i—x^ (i—x^y 

12. w=_^ du^J''^^. 



166 DIFFERENTIAL CALCULUS. 

13. ^={\+x)^\'-^x du=^^^^. 

271— » • 

14. u=.(JT=:x)u-T+^) du= -<^_^'+^^ . 

16. u:=. J_^ ..,,. du=^{ydy-xdx)^ 

16. u=2xja''4-x^ du=^ — ' ^ ■ . 

17. «=(a+j5— i) c?w= . 

^' '. x^ 



18. yz={a+JxY dv= Ha+J^Ydx 

^Jx 

^o 1 jt —Sdx 

19. jr== ,=:^ cfy= — ^:, = — 

(a+Jxy 2jxia+Jxy 

20. P=2xy' dF=2y^dx+4xydt/. 

21. P=_JL_ r^P= — ( 2y^ dx-}-4xydy) 

2xy^' 4ar2y* 

22. g=(J^q:^)(76M=^),^ 

23. Find the differential co-efficient of {a+x^){b-]-3x' ). 

-4w5. 15x*-\-3x^b+6ax, 

K. B. Put «=(a+a;3)(5+3a;2), and find the value of — . 

dx 

23. Find the differential coefficient of a cube whose side is x. 

The function is then u=x^, and du=(3x'^)dx, and Sa;^ is the 
coefficient required, showing that the differential of the side must 
be multiplied by the coefficient Sx^, to obtain the differential of 
the volume, and this will explain the general application of differ- 
ential coeffieients. 




CIRCULAR FUNCTIONS. 157 

CHAPTER II. 
The Difierential of Circular Functions. 

(Art. 7.) Let AB be a circular arc 
designated by x, and let it receive an in- 
crement k, as represented by Bp, and 
from this we are to determine the value of 
(op), (oB), and TH. 

When Bp is so small that the chord and 
the arc may each be considered a right 
line, then h becomes (dx), the differential 
of the arc {op) is the differential of the 
sine X, (oB) is the differential of the co- 
sine Xf and Tffis the differential of the tangent x. 

The two triangles poB and CDB are equiangular and similar. 
Each has a right angle, one at 2), the other at 0. 

The angle pBC is a right angle, so is oBD ; from each to take 
the common angle oB 0, and we shall perceive that the angle 
pBo is equal the angle CBD. Whence the angle 02:)B is equal 
the angle BCI), and the two triangles give the following pro- 
portions : 

pB : Bo :: CB : BD, (1) 

pB : po : : OB : CD, (2) 

If the radius CB is taken equal to uniii/, and pB suflficiently 
small to call it dx, then proportion ( 1 ) becomes 

dx : — d.cos.x : : 1 : sin.a:. (3) 
and (2) becomes dx : dsin.x : : 1 : cos.a:. (4) 

Whence d. 

(A) 



And 



'.cos.a;= — sm.x,dx.) 
dsm.x=cos.xdx, i 



It now remains to find the value of TJI, the differential of 
tan.a;. For this purpose we will resolve the triangle CTHirig- 
onometrically, thus : 

sin. CffT : CT : : sm.IICT : RT. 

Now let it be observed that the sine of the angle CHT is very 
11 



tm DIFFERENTIAL CALCULUS. 

nearly the same as the sine of the angle CTA, which is equal to 
the cosine of the angle TO A or cos. a;. 

Also, as the angle JICT is an extremely small angle, (by hy- 
pothesis, ) and as the sine of a very small arc is the same as the 
arc itself, therefore s'm.IfCT=pB=dx. 
Whence the preceding proportion becomes 

cos.a; : CT : : dx : c?tan.ar. 
Now, in the similar triangles CDB, CAT, we have 
CJ) : CB : : CA : CT. 

That is, cos.a: : 1 : : 1 : CT= ^ 



cos.a; 



Therefore, cos.a; : : : dx : dt2i.n.x, 

cos.a; 

Or c?tan.a;= — — _ 



(Art. 8.) All this can be drawn more readily from the trig- 
onometrical formula, but we gave the preceding article because 
we deemed it essential for a learner to have a geometrical view 
of the subject. 

Now we will show the same as follows : 
Let x= the arc AB, and k= the arc Bp, then by trigonometry 
sin.(a;-{-^)=sin.iCCos./i+cos.a;sin.A. (1) 
cos.(x-\-h)^=cos.x COS.A — sin.a; sin.h. (2) 
Now if we suppose h represents an extremely small arc, then 
008.^=1, and sm.h=h, and (1) becomes 
sin.(a;-|-A) — sm.x=co8.xh. 
But under this supposition the first member of this last equa- 
tion becomes c?.sin.a;, and k=dx, and the equation itself 
becomes 

d. sin.a;=cos.a; dx. (3) 

By parity of reasoning, equation (2) becomes 

cfcos.a:= — sin.xdx. (4) 

We perceive that (3) and (4) are the same as (A) in Art. 7, 
as they ought to be. 



CIRCULAR FUNCTIONS. Ml 

To obtain the diflferential of a tangent we observe that 

, ^ sin.o; 
tan.a;= 

COS. a; 

The differential of the first member of this equation is simply 
(dta.Ji.x), and the second member must be differentiated by the 
rule applicable to fractions. 

Thus ^f.. ^- c^siD.^cos.a;— c^cos.a;sin.a; 

cos.^a; 

Substituting the values of c?sin.a;, and of d cos.a;, taken from (3) 
and (4), we have 

^^^^^_ (<'os.'^+8m.^x)dx _ dx 

COS.^iC COS.^iC 



By trigonometry, secant.a;= (Radius unity.) 

Whence d*aeG.x=- 



Also, cot.ic 

Whence d.cot,x= 



cos.a; 

mn.xdx tan. a; dx 

cos.^ic cos.a; 
1 



tan.ic 
tan.a; — dx 



tan.^o; cos.^artan.^a: 
But tan.a;cos.a;=sin.ar. 

Therefore dcot.x=— . 

sm.^ic 

By trigonometry cosec.a:= — — . 

sm.a; 

fin. i. J — cos.xdx 

Therefore a;.cosec.a?=. 

sm.^a? 

But £??f=_l_. Whence c?cosec.aJ=. ""^ 



sin.ic tan.rc tan.a^sm.a; 

It is obvious that the differential of the versed sine of an arc 
is the same as the differential of the cosine, differing only in 
their signs. 

For the sake of reference we collect the preceding results, 
showing the differential expressions for all trigonometrical lines. 



UO DIFFERENTIAL CALCULUS. 

dsm.x=cos.xdx. (a) dcot.x= — ^ . (e) 



dcos.x=-'Sm.xdx. (b) daeo.x=J^:Rf^. (/) 

d ver. siii.a;=sin.« dx, (c) c? cosec.a;= (a) 

tan. a; sin .a; 

<?tan.a;=_-^-. (cf) 

To diflferentiate any power of a sine or cosine, we proceed as 
follows : 

Let t<=sin,"iP. (1) Put y=Qin.x. (2) 

Then u—^. And du—u'if^^dy. (3) 

But from (2) we find dy-^o.o^.xdx. Now substituting the value 
of y, and dy, in (3), we have 

du=^n%\VL^~'^xQ,o^.xdx. 

From this we perceive that the difiFerential of ««=sin.*a: must 
be <?M=4sin.3a;cos.a:cfe. 

If we have w=cos."a;, a similar process will give 
<;%= — n COS."" 'ar sin.a; dx. 

The practical uses of these equations will be shown in ftiture 
portions of this work. 

(Art. 9.) Hitherto we have shown the differential of sines, 
cosines, tangents, &c. considered as the function of an arc; we 
now propose to show the differential of an arc regarded as a 
function of its sine, cosine, tangent, &c. 

-When we represent a sine by ar, the arc to which it corres- 
ponds is designated thus 

arc(sin.=a;). 

If we represent a cosine by a?, its corresponding arc would be 
designated thus 

arc(cos.=ar;, 
1?^ich is read, an arc whose cosine equals x. 



CIRCULAR FUNCTIONS. 161 

This notation not being satisfactory nor convenient, modern 
mathematicians have adopted the following : 

sin.'^ar, cos."^a?, tan.'*a;, &c. &c. 

Thus M=sin.'*a;, indicates that x is the numerical value of 
a sine, and u is the numerical value of the corresponding arc, 
therefore the equation may be written sin.M=ar. 

Similarly cos.'*a;=z^, is the same as cob.u=x. 



If a; represents the sine of an arc, ^1 — x^ must represent its 

cosine, and _______ must represent the tangent of the same arc. 

J\—x^ 
Take sin.w=a:. 

We differentiate the iBrst member as a sine, and the second 
member as an algebraical quantity ; therefore 

COBM du=dx, 

dx 
Whence du= — ♦ ( 1 ) 

J\—x^ 

Let ^ be a tangent, and u its correspondmg arc to radius unity, 
then we may write 

w=tan.'*^. 
Or tan.w=^. 

Now by equation (c?) of (Art. 8,) we have 

COS.^M 

du=zdt(GOB.^u)=dt = — ^—, (2), because ~, 

is the cosine of an arc when radius is unity, and t the tangent. 
Again, let cos.M=y. 

Then — sin.w du=dy. 

Or du:=^ — ^y, . (3) 

Equations (1), (2), and (3), of this article will be frequently 
referred to and applied in the integral calculus. 



162 DIFFERENTIAL CALCULUS. 

We give tlie following examples to discipline the learner : 

1. Qiven ^m.{mx)=Ui to find du. 

QOB,{mx)d{mx)=idu. 
Or mdxcos.mx=du, 

2. Gfiven u=8mr^ ( ^ Y or sin.w=-— ?-, to find du. 

..,.,^.,^ -^^dx{l+x-^-^xdx{\-x-) 

But if the sine of an arc is , the cosine of the same arc 

l+x^ 

Zx 
is , because sin.2+cos.^ = l. Therefore 

2x , _ '-2xdx( 1 4-a;^ )—2xdx( 1—x^ ) 
1+^ ^ (l+x^y 

2dx 
Whence by reduction du= — — — 
^ 1+x^ 



3. Given u=8mr^J{l — -\ oy sm.u=Jf -\tofind6xL. 



— dx 
Ans, du: 



2^1— ^^^ 

4. 6Hven go8.u=4x^j to find the value of du. 

—X^x^dx 
Ans. du- 



^1— 16a;* 



5. 


Given sin.M=- to find du. 




Ans. au-^^^^. 


S. 


Given 8m,z=2ujl — u^ to find dz. 

Ans. dz=: —. 



Vl— w' 



LOGARITHMS. 163 

CHAPTER III. 

On the Differential of Exponential Quantities and 
liOgarithms. 

(Art. 10.) Hitherto the variable quantities have either been 
algebraic or circular, but we may have an equation in the form 

y=a-. (1) 

In this equation the exponent x is variable, and if it becomes 
(x-\-K) we are to show what effect that will have on the value 
of y. 

As in our previous notation, if a; becomes (x-^-h), let y become 
y\ then 

y'=a»+h=a'^ a*> . (2) 

Subtracting the original equation, we have 

y' — yT=za^a^ — a^ . (3) 

That is, yjll=a^^l. (4) 

Or ^^+l=ah. (6) 

y 

If W3 put a=l-|-5, we can expand the second member of (5) 
by the binomial theorem thus : 

2 3 ' 

This substituted in (5) and one dropped for each member, and 
dividing by h, we shall have 

This last equation is true for all values of A ; it is true then 
when h has a value inconceivably small, but in that case y' — y 



IM DIFFERENTIAL CALCULUS. 

becomes dy, and h becomes dx. On tliis supposition the pre- 
ceding equation becomes 

ydx 2^3 4^6 6 ^ ^ 

The second member of (6) is a series of constant terms, and 
is always the same, while a in equation ( 1 ) remains the same. 
Let the sum of this series be represented by Ay then (6) be- 
comes 

^^Adx, (7) 

y 



If we take A=^ — Then (7) becomes 



dx=m^. (8) 

y 

If we observe equation (1) y=a^, we must recognize a log- 
arithmetic equation, x is the logarithm of the number y, and the 
base of the system is a. 

Equation (8) gives us a general rule to diflferentiate a loga- 
rithm : 

Rule. The differential of a logarithm is equal to the differential 
of the number divided by the number, multiplied by the modulus of 
the system. 

When the base is so taken as to make -4=1, then will ?w=l, 
and we shall have the hyperbolic or Naperian system. For con- 
venience merely. Lord Naper the first investigator of logarithms, 
assumed A=\. This system is still used in mathematical ope- 
rations, and the results changed into the common system, if 
need be, by applying the factor m. When m=i equation (8) 
gives the following rule to differentiate a logarithm : 

Rule. Take the differential of the quantity and divide it by the 
quantity. 

Practical application is made of equation (8) in the author's 
treatise on Surveying and Navigation, and we will give a few 
examples here by way of illustration. 



LOGARITHMS. 166 

1 . The logarithm of 10452 is 4.01 919941, what is the logarithm 
of 10452.12, the modulus of the system being 0.43429448?* 

Here y= 10462, c?y=0.12. 

Therefore dx=:'^'-^L 

10452 

To 4.01919941 

Add <?ar=0.00000498 

Log. 10452.12 =4.01920439 

2. The logarithm of 104521 .2 is 6.01920439, ivhat is the loga- 
rithm of 104520. l'^ 

Here y= 10452 1.2. dg=-^0.7, 

^ . 43429 448(— 0.7) 
10452^2 

To 5.01920439 

Subtract . . . c?ir=— 0.00000028 

Log. 104620.7 =5.01920411 

Thus we might give examples without end. 

(Art. 11.) Logarithms are exponents, therefore the addition 
of two logarithms corresponds to the logarithm of the product 
of the two members. 

Thus log. 2^=log.oa=log.a4-log.a=2log.a. 

The log. of a is half as much as the log. of a^ . 

In the common system the log. of 100 is 2, the square root of 
100 is 10, and its log. 1. The square root of 10 is 3.16227766, 
therefore the log. of this number is 0.50000. Thus we may go 
on extracting square root for succeeding numbers, and halving 
the log. for the corresponding log. 

* We will soon show how this number may be found. 



166 DIFFERENTIAL CALCULUS. 

The following table shows some of these results : 

Numbers. Logaritlims. 

10.00000 1.00000 

3.1622776630 0.60000 

1.7782793430 0.25000 

1.3335214070 0.12600 

1.1547819700 0.06260 

1.0746077770 0.03125 

1.0366328673 0.015625 

1.0181521828 0.0078126 

1.0090352733 0.00390625 

1.0045074297 0.001953125 

1 .002251 1 809 0.0009765625 

1.0011249572 0.00048828125 

1.0005623151 0.000244140625 

1.0002811174 0.0001220703125 

1.0001405488 0.00006103515625 

1.0000702719 0.000030517578125 

1.0000351353 0.0000152587890625 

1.00001756752. 0.00000762939453125 

Thus we might go on, but we hare gone far enough to illus- 
trate the possibility of finding the value of m independently of the 
Naperian system. 

The log. of 1 is in every system. Our last log. just found 
corresponds to a number a little greater than 1, but the decimal 
is so small in relation to 1, that it may be taken for the differ- 
ential of 1 . 

Equation (8) gives us dx=m^. Whence m=i- . 

y dy 

Butif?/=1, cfy=0.00001 756752, and c?a;= the last log. 

rrs. 0.00000762939453125 ^ .^^nn^ i 

Then m= = 0.434294-4-. 

0.00001756762 ~ 

To fix these principles in mind we give the following exam- 
ples to differentiate. The word logarithm is indicated by log., 
and indicates the hyperbolic or Naperian log. unless otherwise 
expressed. 



LOGARITHMS. 167 

I. Oiven u=\og.( ^ ~\ tojind dvL. 



X 

We first take tlie diflferential of the second member as an al- 
gebraic quantity, thus : 






a^+x 



a^dx 



Whence c?w= X 



Ja^-\-x^ =, d^'dx , ^^, 



(a^+x^)Ja^^x^ x x{w'+x^) 



2. Oiven u=\og.{x-{-Ji-\-x^) to find 6m, 

dx 
Ans. du=^-^==r-' 
J\+x- 



3. Given u=-.-^:^\og.(xJ— \-\- J I— -x^), to find du. 
V-1 



N. B. Put a=J — 1 for the sake of perspicuity. 

dx 



Ans, du=. 



4. Given n=log.(^l—x^) to find du. 

Ans. du= 

(l-x-) 

5. Given u=\og,{Sx^ -^-x) to find du. 

Ans. du=( ?^i:L\dx. 

\ 3x--^-x/ 

N. B. We can if we please use logarithms to differentiate 
common algebraic quantities. To show this, we take example 
11 from (Art. 6,) of this volume. 

6. Given u=—L — , to find du. 

1 — x'^ 

Take the log. of each member, then 

log.M=log.(l+a;2) — log.(l— aj2). 



168 DIFFERE>fTIAL CALCULUS. 

Now differentiate each member by the logaritbmetic rule, and 
we find 

du ^xdx . Sixdx 4xdx 

Tt 1 _L_/..2~ ' ' 



n J 4xdx . . l-\-x^ 4xdx j. 

7. ^ve« u= to find ^u. (From Art. 6. ) 

\og.u=n log.ar — w log. ( 1 +a;) .• 
du ndx ndx ndx 



X (1+^) x(l-\-x) 



rdx / a;" \ na^'^dx , 

I \(\A.xY ) ~7--~--^- 



aj(l+a;)\(l+ar)»/ (1+a;)"** 



8. Oiven u^^xja^-^x^ to firtd diu, making use of logarithms. 

An.. du=(Ji±^£J±^, 
Ja^+x^ 



9. Cfiven u=\og.(2xJa^-\-x'^) to find du. 

The differential of the quantity under the log. was found in 

the last example, hence the answer to this is found simply by 

dividing that answer by 2xja^-^x^. 

. , (a^+2xndx 
Ans, du= \ \ \ .. 
x[a^-\-x^) 

The following examples come nearer the practical uses of these 
principles : 
(Art. 12.) 

10. Given w=log. sin.a; to find du ; that is, given the log. sine 
of any arc to find its differential , or its rate of increase or decrease 
at that point. 

y cos.xdx . J 

au= — =cot..'r dx, 

sin. a; 

This result corresponds to the modulus of unity : for the modu- 
lus of our common system we must multiply by 0.43429448=?w. 



LOGARITHMS. 160 

For example, if we assume a;=25°, and also assume dx=\', 
the diflferential, or the diflference between the log. sine of 26° 
and the log. sine of 26° T is expressed by wcot.26°Xl'. 

Log.m —1.637784 

cot. 25° 0.331327 

Log. sine 1', less 10 —4.463726 

.0002709 —4.432837 

To the log. sine of 26° 9.625948 

Add the differential. 000271 

Log. sine of 26° 1 '= ..9.626219 

We might assume dx==2' as well as 1', without error as far as 
six places of decimals ; but it would not do to assume dx= any 
large number of minutes ; hence the differential calculus must 
be applied with judgment. 

1 1 . CHvm n=\og. cos.ic to find du. 

Ans, du= — tan.xdx. 

To apply this, we demand the variation of the log. cosine of 
34°, corresponding to T increase of arc. 

Log. m —1.637784 

tan. 34°, (less 10) —1.828987 

Log. sine 1', (less 10) —4.463726 

c?w=0.00008521 —5.930497 

To log. COS. 34° 9.918574 

Subtract du 0.000085 

Log. COS. 34° r .9.918489 

12. CHven u=log. i2Lii.x to find du. 

, dx dx 2dx 

du= = =-; 

cos.^ajtan.ic cos.irsm.ic sm.2a; 

What is the variation corresponding to T to the logarithmic 
tangent of 40°? 



*70 DIFFERENTIAL CALCULUS. 

Log. m —1.637784 

Log. sin. r+log.2 —4.764756 

Sin.2a;=sin.80° complement 006649 

</m=0.00025653. — 4.409189 

To log. tan. 40° 9.923813 

Add du 267 

Log. tan. 40° r .9.924070 

13. Oiven w=log.(cot.a:) to find du. 

^ — dx — dx — 9.dx 

sin.2a;cot.a; cos.ajsin.ir sin.2a? 
This last result shows that the log. tangent and log. cotangent 
vary alike in amount, the first positive, the last negative. 

(Art. 13.) The use of logarithms is very essential in differ- 
entiating examples like the following : 

It is customary to represent the base of the Naperian system 
by e, and the log. of the base of any system is 1 ; hence, if we 
have any equation in the form u=e^ and take the log. of each 
member, we shall have \og.u=x simply, and if u=e^i/f log.tt= 
2?+log.y, &c. (fee. 

14. 6Hven u=e^(x — 1) to find the value of du. 

Log.«^=ia;-j-log.(a; — 1). 

du J . dx xdx 

— z=2dx-\- = . 

u x — 1 X — 1 

du=^^=.e-xdx, 
x—l 

16. Given u=e''{x^ — ^x^-\Sx — 6) to find d.VL, 

Log. w=a:4.1og. (a;^ — 3a; 2 +6a?— 6) . 

df^^^^ . (3a;^— 6a;+6) dx _ x^dx 

u ^"''aj^—Sa;^ J^Qx—Q~ x^—3x^ +6ar— 6* 

—= ^- =e^x^. Or du=e^x^dz, 

dx a;3— 3«2+6a;— 6 



LOGARITHMS. 



16. Given u= to find dn. 

1 — X 



17. Given u=e^\og.x to Jind du. 

Ans. du=( — pI ' je^dx. 



18. Given u= to find du. 



Am,du=J!^-, 

(e-+in 



19. Given u^x'^ilog.xy to Jlnd —. 

dx 

Put z=:log.x, then u^=x^^. 

Ans. _=(mlog.a;+w)a;"-»(log.a;)*-». 

20. Given t.= l*il2i:^~^^M+^ ^o/;^^ du. 

4 8 ^32 

21. (Tjww «=a?y to find the value of du. 

Log.«<=ylog.ar. 

^=log.^.iy+3,^. 
du=xy \og.xdi/-\-yxy~^dx. 



22. G'iwTi «=log.(cos.a?+^ — Isin.a;) to find du. 

Ans. du=J — Idx. 

23. CHven «=!2i£ to find du. 



^n.. cfe.= (lzi^-^)rf:r. 



1P8. DIFFERENTIAL CALCULUS. 

CHAPTER IV. 

Successive Differentials. 

TAYLOR'S THEOREM. 

(Art. 13.) When we Iiave an equation u=^f(x)^ its differen- 
tial coefficient is generally another function of Xy symbolically 
represented thus : 

dU J. r >. 

This new function of x can again be differentiated and divided 
by dx, giving still another function of x, then we shall have 

until the last differential becomes constant or valueless, as the 
case may be. 

For example, let u^ssx^. 



The 1st diff. 


coefficient 


is 


^=30:^ 
dx 


The 2d is 






dx^ 


The 3d is 






d^U^Q 

dx^ 



Here the operation must stop, as the second member is con- 
stant. By this we perceive that if u=x'^ after n differentiations, 
the second member will become constant and terminate the ope- 
ration. 

We sometimes write 

du dp <^9 A. 

dx dx dx 

Then will — =i>, =g, =?*, &c. 

dx dx^ dx^ 

du is the differential of u ; 

d^u is the second differential ; 

d^u is the third differential ; &c. 



SUCCESSIVE DIFFERENTIALS. 173 

It should be remembered, that the exponent which accompa- 
nies the characteristic d^ indicates the repetition of an operation, 
and not a power of the letter c?, which is never considered as a 
quantity, but merely as a sign. 

The expression dx^ signifies the square of dx, or (dxY , and 
not the differential of a;^, which is usually denoted by d,x^ or 
d{^x^)\ again, dx^ signifies the cube of dx or {dxY y and so on. 

(Art. 14.) If we have a function of the sum or differeme of 
two variables, the differential coefficient will he the saine whichever be 
supposed to vary, the other being constant. 

For example, let u^=^(x-±iy)^ . 

Take the difierential coefficient on the supposition that x is 
variable and y constant, then 

^=i(x±y)K (1) 

Now on the supposition that y is variable and x constant, and 

dy 

Comparing equations (1) and (2), we perceive that 

du du 

dx dy 

Taking the differential coefficient of (1) in relation to ar, and 

d^u 

dx' 
And the differential coefficient of (2) in relation to y, is 

g=12(.±y)'. (4) 

Comparing (3) and (4), we perceive that 
d^^d'^u 
Tx^ If 

Thus we might show that 

d^u__d^u and d^u __d^u 



-^-12(a:±y)^ (3) 



174 



DIFFERENTIAL CALCULUS. 



(Art. 15.) If we have u=^fx and suppose that a: becomes 
(a;+A), and in consequence of this u becomes %', then 
u'=:^u-\-Ah\Bh-'-\^Ch^^Bh^, &c. (1) 

To give the learner a clear comprehension of this, we will 
illustrate it by one or two particular examples. 

Let w=aa;*. 

Now suppose X becomes x-^-h, and u becomes u\ then 
u'=a(x+h)^=ax^+4ax^h-\-6ax^h''-\-4axh''+ah^, (2) 

Now it is visible that the first term of the second member is 
Uf and if we -pvit4ax^=A, Qax^=Bf &c. we have 

u=::u-\-Ah-{-Bh^-\-Ch^-\-Dh'^f &c. which we proposed to show. 
Again, let u=.a-\-bx-\-cx^ ^ and suppose x becomes x-\-hf &c. 
Then u' =a-\-b{x-\-h)+c{x-\-hY . 

Or u'z={a-\-hx-\-cx^)^(b-\-2cx-\'Ch)h. 

Here again we have 

u'=:^u+Ah+Bh^-\-Ch^+ &c. 
according to the degree of the variable x in the original function - 
Resume the equation u=ax'^, and take its successive differ- 
ential coelficients thus : 
du 
dx 
d^u 
dx 
d^u 



=4ax^=A 
= 12ax^=2B. 



=24aar=2.3.(7. 



dx' 

=24a: 

dx^ 



:2.3.4i>. 



Comparing these results 
with equation (2). 



Substituting these values of A, B, C, &c. inequation (1), we 
have 



^dx ^dx^ 2 



d^u h^.d'^u A* 
'dx^ 273 dx^ 2.3.4' 



&c. 



a general expression for the development f(x-\-h)=u\ and this 
is Taylor's Theorem, from Dr. Brook Taylor, an English mathe- 
matician who discovered it about the year 1715. 



SUCCESSIVE DIFFERENTIALS 175 

(Art. 16.) In the preceding article we drew out Taylor's 
theorem by inspection. Let that be well understood, and the 
learner is prepared to appreciate the following general demon- 
stration : 

Let u=f{x). 

Now suppose X to take an increase y, and in consequence of 
this, u becomes w', then 

u'=^f{x+y)=^u+Ay+By^+Gy^+Dy\ &c. (1) 

In the second member, u, A, B, <fec. contain functions of x, 
and the constants that enter into x. 

Take the differential of each member in relation to a; as a 
variable, and divide each term by dx, then we shall have 

du' du.dA ydB ^ l ^^ 31^-^ 4_l_ r?r (9\ 

dx dx dx dx dx dx 

Take the differential coefficient of (1) again, regarding a; as 
constant and y variable, and we shall have 

^'^'=^-f2%+3(7y'+4Z>3/3+ &c. (3) 
dy 

Now the first members of (2) and (3) are equal by (Art. 14), 
therefore the second members are equal, and the terms contain- 
ing like powers of y are equal. (Algebra Art. 128.) 

Therefore ^=^, ^^=25, 4^=3(7, 4--=4A &c. 
dx dx dx dx 

T> du . d^u dA ^r) d^u dB ^^ « 

Because — =A, =_=2^, = — =3(7, <bc. 

dx dx^ dx 2dx^ dx 

Whence ^=^, ^=^, C=_i!-^_, i)= ^'^ ^ 



dx dx^2 dx^ 2.3 dx^ 2.3.4' 

Substituting these values of A, B, C, &c. in (1), we have 

, . du . d^u y^ . d^u y^ , d'^u y^ . . 
^dx^dx'' 2^dx^ 2.3 ' dx* 2.3.4 ' 

NT> rm, • du d^U d^U « ,r 

. JtJ. The expressions u, — , , , (fee. are the same 

dx dx"^ dx^ 

as X, X\ X", X"\ (fee. in Robinson's Algebra, (Art. 171,) 

page 273, and are there called derived polynomials. 



176 DIFFERENTIAL CALCULUS. 

MACLAURIN'S THEOREM. 

(Art. 17.) Maclaurin, a Scotch matliematician, has given us 
a theorem very similar to that of Taylor, which demonstrates 
the binomial theorem, and enables us to develop any function of 
a single variable, provided the development is susceptible of con- 
taining the ascending powers of the variable. 

But the theorem does not apply to other forms of development. 

Let u={a+xY={a)-\-Bx+Cx''+Dx''-\-Ex\ &c. (1) 

Here we are sure the first term of the development (a) does 
not contain a?, and B, (7, i?, &c. are each independent of the 
value of X. 

Take the successive differential coefficients of (1), and the re- 
sults will be as follows ; 

^=^B+2Cx+Wx^-\-^Ex^-\' &c. (2) 
dx 

^=:2C+2.Wx+3.4Ex^+ &c. (3) 
dx^ 

:5?!^==2.3i>+2.3.4^a;+ &c. (4) 

cix 

Equations (1), (2), (3), &c. are all true for all values of x; 

they are therefore true when x=0. Making this supposition, 

they become 

u=a*=:(a) 



C= 



dx 



dx^2 



jD=J^-. &C. 

dx^2,3 

Substituting these values of B, (7, i), &c. in (1) and we have 

^ ^^\dx/ ^Kdx''/ 2^\dx^/2.3^ 
The first term (a) is whatever u becomes when the variable 
is made equal to in the primitive function. 



SUCCESSIVE DIFFERENTIALS. 177 

APPLICATION. 

Let us now apply this theorem to this very example ; that is, 
develop (a-j-^y by it. 

g=«(»-i)(<»+«r'. 

^=«(»-l)(»-2)(«+^)-». 

^=n(n—lXn^2){n—3)(a+x)'^', &c. 
dx* 

Making x=0, these equations become 

dx dx^ dx^ 

Hence, by substituting these values in the formula, we obtain 



<^)C-i^)"-'^'+ ^- 



This is the same result as would arise from the direct applica- 
tion of the binomial theorem, and this formula can be used to 
develop binomials generally — but there would be no advantage 
in using it for common cases, for the direct application of the 
binomial theorem is less circuitous and more brief. 

But this theorem is more powerful than the binomial theorem, 
and will apply to other functions as well as to simple algebraic 
binomials, — hence its utility. 

To show the power of this theorem, and at the same time draw 
out useful mathematical truths, we give the following 

EXAMPLES. 

1 , Develop a^ into a series containing the ascending powers of 
X, if possible. 

Making a;=0 the function becomes 1, a rational finite quantity 



178 DIFFERENTIAL CALCULUS. 

therefore the development is possible — as the demonstration was. 
general under that hj^pothesis, 

In equation (7), (Art. 10,) we find -^==^a*. As u4 is a 
constant quantity, and e<=a^, a second differentiation will give 

A third ^=zA^a^, 

dx^ 

A fourth ^=:^4^x. <fec. 

Making ar=0 in these equations, as in the previous example, 
and we have 

^=A, ^=^», i'Ji^A^. &c. &c. 
dx dx^ dx^ 

Therefore equation {I ) becomes 

a^ = l+^.+d!^V^^V---+-^i^+ &c. 
' 2 ' 2.3 ^2.3.4^2.3.4.5 ' 

As X is unrestricted in value, we may make x=- 

Then a^=l+l+i+J_+_i_+ ^ + &c.=f. 

' 2 ' 2.3 ' 2.3.4 ' 2.3.4.5 

The value of e taken to seven decimal places, it is 2.7182818. 
This is the base of the Naperian system of logarithms, and it 
is much used in analysis. 

From the last equation we find a=e . 
Taking the logarithms we have log.a=-4 log.e. 

Or ^=l5Sf. 

log.e 

Now since a and e are known, A is known. If log.a=I, 

A=^ , and if log.c=l, -4=log.a. That is, A is equal to 

log.e 



SUCCESSIVE DIFFERENTIALS. 179 

the Naperian logarithm of 10. We shall soon discover that the 
value of this logarithm is 2.302585093. 

The modulus of the common system of logarithms is the 
reciprocal of A designated by m in (Art. 10); therefore ?n= 
. 434294482, corresponding to the result approximately obtained 
in (Art. 11). 

(Art. 18.) To show the distinction between the theorems of 
Taylor and Maclaurin, we will now apply the former to the 
development of this same function 

Let u=a^. (1) 

Then u'=a^*^ (2) 

And by the theorem 

u=U'\- — h-\- -4- + (fee. (3) 

^dx ^dx^ 2 ^dx^ 2.3^ ^ '^ 

Taking the successive differential coefficient of ( 1 ) we find 

dx dx^ dx^ 

Substituting these quantities in the formula, and 

2 ' 2.3 ' 2.3.4 
Divide each side by a ^ , and 

' ' 2 ' 2.3 ' 2.3.4 ' 

As h is arbitrary, we may put h= — , then 

Al 
\_ 

a^=l+l+l+_l_+__l_, &c. =e. 
2 2.3 ' 2.3.4 

Or a=e , as before. 

In Maclaurin's theorem the differential coefficients i — V 

\dxr 

(d^u\ 
-— - \, &c. correspond to the variable x=0 in the second mem- 
ber, and they are put in parenthesis to distinguish this theorem 
from that of Taylor. 



180 DIFFERENTIAL CALCULUS. 



2. For a secmd example, let u=sm.x, and w'=sm.(a;-|-^). 

du d^u . d^u 

—= cos.a:, =— sin.o;, -— 

dx dx^ dx^ 



Then(Art. 7,) ^=cos.a:, ^!^=— sin.o;, ^=— cos. ar. 



6?*« . d^u B 

— =sin.a;, , — =cos.ar, &c. 
dx* \dx^ 

Substituting these values in Taylor's formula, we find 
sm.(rr+A)=sin.a;+cos.a;_ — sm.a; — cos.ar -]-sin.a; 



1 1.2 2.3 2.3.4* 

This is true f6r all values of x\ — it is true then when a;=0, 
and this supposition gives sin.a;=0, and cos.a;=l, and the re- 
sult becomes 

sm.A=A — 4- — . + (KC. 

2.3 '^2.3.4.5 2.3.4.6.6.7 ' 

Rbmakk.— This operation compared with that in Geometry, pages 221, 
222, 223, for the same object, shows the superior power of the calculus 
over common geometry in a very clear light. 

3. For a third example, let u-=cos>.x, then u'=Q,0B.(^x-\-h). 
Taking u, and the successive differential coefficients of u, and 
substituting them in the formula, we shall find 

cos.A=l — — 4- — -I- &c. 

2 ' 2.3.4 2.3.4.5.6 

These results may also be deduced from the theorem of Mac - 
laurin. 

These formula are used to compute the sines and cosines of 
small arcs when the arcs are known. 

(Art. 19.) By Taylor's theorem we can easily develop a 
logarithmic function into a series. 
Let «=log.rr. 

u' :=\og.{x-\-h\ 
Taking the successive difi'erential coefficients of u, we find 
dM^\ d^u___l d^u_2 

dx Ic ^~" ^' ^3""^' 

^____3 d^u_S.4 

^ x^* dx^ x^' 

on the supposition that the modulus is unity. 



SUCCESSIVE DIFFERENTIALS. 189 

These values substituted in the formula give 

&^TJ ^ ^a; 2x^^3x^ 4x'^5x^ ^^ 

If h be made minus, the result of the formula will be 

log. (a; — h)z=\ocf,x—- — — - — — — — — &c. (2) 

Subtracting the latter from the former, we have 

log.(a:4-^)— log. (a;— A)=_+ + + + <fec. 
x 3^3 bx^ Ix^ 

If we make a;+^=2 and x — A=l, then a;=f, and ^=^. 
Also _=-, and (3) becomes 

X o 

log.2=2(^^+l JL+LJ_+i. JL+)&c, 
^ \3 ' 3 27^5 243 ' 7 2187 ' / 

This gives the hyperbolic logarithm of 2=0.69, and so on. 

As formula (3) is not convenient for all numbers, we will 
modify it. It is obvious that the first member is greater than 
1, therefore we may assume 

This ffives _= , and these values substituted in the 

^ X 2^+1 

formula, give 

^\ z J \(23+l)^ 3(22+1)3^ 5(22+1)5^ / 

Or iog.(2:+l)=log.2+2(^_i_+ 1 + ^ +\ 

B\-T J S -r V 22:+1^3( 2^+1)3^ 6(22+1)5 V 

This formula gives the hyperbolic logarithm of {z-\-\) when 
the log. of z is known. 

When z=\, log.2=0, and the formula gives the hyperbolic 
log. of 2. Because 2^=8, three times the log. of 2 will give 
the hyperbolic log. of 8. 



182 DIFFERENTIAL CALCULUS. 

Now making z=8, an application of the equation will give 
the hyperbolic log. of 9. Then making 2=9, another applica- 
tion will give the hyperbolic log. of 10, which is 2.302585093, 
and it is represented by A, or log.a in (Art. 17.) 

In (Art. 10,) we represented the modulus of the common sys- 
tem by m, and m= — 

Hence m= ^ =0.43429448. 

2.302585093 

Therefore the formula for common logarithms is 

log(^+l)=log.0+.86858896 ( ^ + + \ 

b\-r J 6 -r \ 2^+1^3(20+1)3^5(2^+1)5/ 

+ &c. 

ANOTHER METHOD OF DEVELOPING LOGARITHMIC AND CIRCULAR 
FUNCTIONS. 

(Art. 20.) We can best illustrate this method by taking the 
preceding example, and comparing the results step by step. 
Let u'=\og.{x-{-h). 

Conceive x to be variable and k constant, then 
du'_ 1 _ 1 
dx ic+A h-{-x 
The second member of this equation can be developed by 
division, or by the binomial theorem. When so developed, we 
shall have 

^=1—^+^-^+^— &c. (1) 
dx h h^^h'^ h^ ' h^ 

This equation shows us that u' expands into ^ series contain- 
ing all the ascending powers of x, and possibly there is a term 
not containing the variable x. 

Therefore we may assume 

u'=AJ^Bx-\-Cx^^Dx^+ &c. (2) 

From this equation we find 

^-^.=B+^Cx-\-^Dx^+^Ex\ &c. (3) 

dx 



SUCCESSIVE DIFFERENTIALS. 183 

The first members of (1) and (3) are the same, therefore the 
second members are equal, and the terms containing equal pow- 
ers of X are equal. Therefore 

B=.l, (7=^_i-, i)=_l-, J^=— _J_, &c. 
h 2^2 _ 3A3 * 4^4 

These values of B, C, Z>, (fee. substituted in (2) give 

o V -r / -r^ 2A2^3A3 4A4^6A5 ^ ^ 

This equation must be true for all values of x. It must be 
true then when x=0. Making that supposition, and 

Substituting this value of A in (4), and the development is 
complete. 

\og.(x4-h)=^\og.h-{.-——Jr~~—~^ (fee. (6) 

This equation is the same as (1) in (Art. 19,) when we change 

X to ht and h to x. This arose from our expanding in place 

h-\-x 

otJ-. 

x-{-h 

Putting u'=\og.{x-\-h), and the result of the operation will 
give a similar equation to (2) of (Art. 19.) 

This principle of operation is best adapted to the development 
of circular functions. 

EXAMPLES. 

Let X represent an arc of a circle whose radius is unity, and y 
its sine. Then the cosine must be 



1 . It is required to find the value of the arc in ienns of its sine. 
sin.a?=y. 
Whence cos.xdx=di/. 



And 



dx__ 1 _ 1 
di/ cos.a; Jl—y^ 



184 DIFFERENTIAL CALCULUS. 



J\—y^ 2 2.4 2.4.6 2.4.6.8 

«?y ~2~2.4~2.4.6~2.4.6.8' ^ ^ 

Here we perceive that dx develops into a series containing the 
even powers of y, therefore x itself must develop into a series 
containing the odd powers of y. As each term containing an odd 
power of J will have the power of y diminished by unify after dif- 
ferentiation, therefore we may assume 

x=Ay+By^+Cy^+Dy''+By\ &c. (2) 

fir 

Whence ~=A-{-SBy^+5Cj/^+7Dy^+9£:y\ &c. (3) 
dy 

Comparing (1) and (3), we find 

2.3 2.4.6 2.4.6.7 2.4.6.8.9 

These values of A, B, C, D, &c. substituted in (2), give 

^~2. 3^2. 4. 5^2. 4. 6. 7 ' 2.4.6.8.9 ^ ^ 

the development required. Knowing the sine y of any defi- 
nite arc, this equation will give the value of x, the arc itself, to 
any required degree of exactness. 

When the arc is 30°, the sine is ^, and this value given to y 
in the equation, there results 

Arc of 30° = 

1 , 1 . 3 ■ 3^6 , 3.5/7 , .^ 

2 ""2. 3. 8*^2. 4. 5732 ""2. 4.6.7. 128'^2. 4. 6.8. 9. 512"*" 

Taking the sum of seven terms, we find 

Arc of 30°=0.523597 

And multiplying by 6, 180°=rt=3.14159. . . . 

2. It is required to find the value of an arc of a circle in terms 
of its cosine. 

Let x=. the arc and z its cosine. 
That is, cos.a;=0. 



SUCCESSIVE DIFFERENTIALS. M6 

dx___ 1 ___ 1 
dz sin. a; /J ^2 

This is the same form as the former example, except the sign. 
Hence the development must be the same as (8) of the first ex- 
ample, except changing y to 2 and changing signs. 
,- In equation (8) the arc and its sine commence at the same 
point and increase together from to 90°; hence, when we make 
ar=0 in (8), y becomes also, and both sides of the equation 
vanish together and the equation is complete. 

Not so with the cosine, for when the arc increases the cosine 
decreases, and when the arc is 0, the cosine is radius or 1 . 

Therefore we cannot develop an arc in terms of its cosine 
independent of the corresponding sine. 

If z is the cosine of the arc a:, it must be the sine of the arc 
(90° — x). Now by example 1, equation (8), the 

Arc (90°— a:)=2+il-+-^-+-l:^iL+ &c. 
^ ' '2.3 2.4.5 ' 2.4.6.7 ' 

Transposing 90°, and changing signs, we have 

Arc a;=arc 90° — z— — — — — '- -— <fec. 

2.3 2.4.5 2.4.6.7 

Here the arc x is developed in terms of the cosine z, as 
required, but the result necessarily includes the arc of 90°, and 
the value of this depends on the development of the sine. 

To find the value of the arc of 90°, we again turn to equation 
(8), making y=l; then that equation becomes 

Arc 90°= !+-!-+_-? 1- &c. 

^2.3^ 2.4.5 "^ 

and this value of the arc of 90° substituted in the preceding 
equation, and we shall have the value of x complete, as was 
required. 

3. Let X he an arc and t its corresponding tangent; required the 
value of X in terms of t. 

tan .ar=/. 
Then .Jl-^dt. 



186 DIFFERENTIAL CALCULUS. 

Or — = cos.^a;. 

dt 

Now we can develop this by Maclaurin's theorem, or as follows: 
In any arc we have the following proportion : 

coQ.x : 1 : : 1 : sec.a;, or cos.«= r. 



sec .a; ^ 

Whence cos.2ar= — ? — = — ? — =Cl4-^2)-i. 

sec.^a; \+e ^ ^ ^ 

Therefore ^={\J{.t^y^=zl—t^J^t^—t^J^t\ &c. (1) 

Here we perceive that the second member contains only the 
even powers of t, therefore the development of x before differ- 
entiation must contain only the odd powers of t. Conse- 
quently we will assume 

x=zAt-\-Bt^+Ct^+Df + &c. (2) 

^-=A+^Be+bCt^+nDt^-\-^Et\ &c. (3) 

Comparing ( 1 ) and (3) we find that 

A=\, B=-x, (7=i i)=— 1, E=:h &c. 
These values of A, B, C, &c. substituted in (2), give 

3^6 7^9 11 ^ ^ 

This formula will give the arc x, provided we know t, any 
corresponding tangent. We learn in geometry that the tangent 
of 45° is equal to the radius =1, and the tangent of 30° is equal 

to — -y therefore 

arc46''=l— JH-t- l-H— tV. &o. 

=(i-J)+(i-4)+(t-,S)+(,V—'T). &c. 

a Q Q Q Q 

=_l_+_:L+_A-+_A_+__±_, &c. 
1.3 ' 5.7 ' 9.11 ' 13.15 ' 17.19 

But the series is not sufficiently convergent to be satisfactory, 
and therefore we will take the value of t corresponding to 30°. 



SUCCESSIVE DIFFERENTIALS. 187 

That is ^=_L, «3=_J^, &c. 

arc 30°=-L— — l_+-_i—,— _-i-^+ 

73 3.3^3 5.3V3 7.3y3 

9.3V3 11.3V3 

/3V 3.3 ' 6.3^ 7.43*9.34 11.3^ 13.36 } 

To sum up this series we take the first term in parenthesis, 1, 
and divide it by 3, that quotient again by 3, and so on; this will 
give us a series of decimals, the second of which divided by 3, 
the third by 5, the fourth by 7, &c. and we shall have the series 
of terms within the vinculum. 

4- Terms. — Terms. 

1 .000000000 1 .000000000 
3) .333333333( 111111111 

6) 111111111( .022222222 

7) 37037037( 005291005 

9) 12345679( 1371742 

11) 4115226( 374111 

13) 1371742( 105518 

15) 457247( 30483 

17) 152416( 8965 

19) 50805( 2673 

21) 16935( 806 

23) 5645( 245 

25) 1881( 75 

27) 627( 23 

29) 209( 7 

1.023709235 .116809651 
.116809651 

.906899584 



1B8 DIFFERENTIAL CALCULUS. 

arc 30°=I?5^9^=I^^^?^==.5235987. 
^3 1.7320308 

arc 30°.6=arc 180°=;t=. 5235987. 6=3.1415922. 

The utility of the calculus will be apparent on comparing this 
operation and its result with the like problem in common 
geometry. , 

REMARKS ON MACLAURIN*S THEOREM. 

The learner must bear in mind that Maclaurin's theorem will 
only apply to such functions as expand into a series according to 
the ascending powers of the variable. Hence, when we attempt 
to apply it to other functions, and it fails to produce the desired 
result, the failure should not be called a failure of the theorem. 

For example : Suppose we have the function ( a-\-~ j , and 

attempt to expand it into a series by Maclaurin's theorem, we 
should fail to produce the proper result because this function 
does not expand according to the ascending powers of the va- 
riable X. It expands in the form A-\-Bx~ ' + Cx'^-^- <fec., which 
is a series containing the descending powers of the variable, and 
the formula was not framed to meet such cases. 

This formula requires that the variable, in the primitive func- 
tion and in the second differential equations, be made equal to 
0, and produce finite results. 

But if we make x=0 in the function ( a-|-- J we shall have 

f a-}-_ j , a result mathematically infinite, and we shall have 

the same indefinite and incomprehensible result in each of the 
differential equations under the same hypothesis. 

We can, however, expand the function («+-) ^J * modi- 
fication of Maclaurin's theorem, which is to make x infinite where 
thai theorem requires us to make a;=Q. 

1 / 1\ " 

Make 2/=-, then ( a-}-- ) becomes (a-|-y)"» and this can be 
X \ x/ 



SUCCESSIVE DIFFERENTIALS. 189 

expanded by Maclaurin's series, making y=0 for the jSrst term, 
but y=0 is the same as making x infinite, because y=~. 

X 

This, and other artifices may apply to other functions, and m 
short, these remarks are made to show the learner that he must 
rely on his judgment in the application of this theorem. 

REMARKS ON TAYLOR'S THEOREM. 

Taylor's theorem is designed to apply to the development of 
a function, whatever value be assigned to the variable. But 
there are some functions which change their form •when some par- 
tictdar value is given to the variable. To such functions the 
theorem will not apply when that particular value is taken, 
but for all other values of the variable the theorem will apply. 
We illustrate this by the following example ; 

Let u=Ja-\-x. (1) 

Then u=iJa-^x-\-y. (2) 

Here x is the variable and y the increment. 

, du I d^u v^ , t, . 1 » 
u —u-\'—-y-\ — ±-.-U &c. IS the formula. 
dx dx^ 2 



1 1 



u'=z Ja-\-x + y— y^ , &c. (3) 

2ja+x 8(a+a;)f 

Now let x= — a, and this development becomes 

Vy=o+ — ^y— ;y^ &c. 

2 JO 8(0)1 

Here the finite quantity Jy is equal to a series consisting of 
mathematical infinites, alternately plus and minus, which is 
indeterminate, if not absurd. Hence, for this partictdar value of x 
the theorem is said to fail, but for all other values of x the 
development is rational and true. 

Now in (2) make (a-\-x)^=A, then u'=JA-\-y. 

The second member expanded by the binomial theorem, gives 

2A' SA' 



190 DIFFERENTIAL CALCULUS. 

Now on the supposition that x= — a, A becomes 0, and 

2(0)2 8(0)"^ 

Here we observe that whatever value is given to the variable y, 
Taylor's theorem, and the binomial theorem, will give the same 
result. 

Hence, when one of these theorems /<7i7 the other fails, but we 
never apply the word fail to the binomial theorem, and it is not 
clear to us that such an expression should ever be applied to 
Taylor's theorem. 

The failure is in the hypothesis and not in the theorems. 
In the present example the hypothesis that x equals minus a 
destroys the binomial form of the function J{0'-\-^)-\-y, and 
makes it Jy sl monomial, and Taylor's theorem is not designed 
to apply to a monomial. 



CHAPTER V. 

The general development of functions containing 
turo or more variables. 

(Art. 21.) We have thus for examined the development of 
functions containing only one independent variable ; it is now 
proposed to extend the same principles to any number of inde- 
pendent variables. 

Let u=f(x,y), 

and if x and y are entirely independent of each other, y may be 
regarded (for a moment) as constant, and then if x becomes 
(x-^-h), Taylor's theorem gives 

f{x+h.y)=u+ h-{- , o"t" ^ 3 o Q + ^^' (^) 

dx dx^ 1.2 dx^ 2.3 

Now if we suppose y to become (y+^), every term of the 
second member of (1) must receive an increment. 



SEVERAL VARIABLE FUNCTIONS. 191 

That is, u becomes 

^dy ^dy^ 1.2^ dy^ 2.3^ ^ ^ 

^ becomes — +_1^^+__:^J^{___+ <fec. 
c?a; dx dy - 

Or — becomes '^^^lc-\-J^L^-2L-+ &c. (3) 
rfa: dx^dxdy ^ dxdy^ 1.2^ ^ ^ 

In the same manner we find that 

^becomes ^!f+_^!ii-;fc+_i^-i!-+ &c. (4) 
dx"^ dx^^ dx^dy ^ dx^dy^ 1.2^ ^ ^ 

becomes — — +— — r— ^i- <kc. (6) 




da:^ dx^ dx^dy 

The developments in (2), (3), (4), and (5), substituted in 
the second member of ( 1 ) will give the following result, which 
is the development of the second state of a function containing 
two variables : 

j\-r >y-r J -r^y -r^^a 1.2 ^^/^ 3 1.2.3^ 



+ duj , d^u ,, , d^u hJc^ . „ 
dx~dxdy ^ dxdy^ 1.2 ~ 

' dx^ 1.2 ' dx^dy 1.2 ' 



(6) 



d^u h^ , s 

-|- <fcc. 

dx^ 1.2.3 "^ 

This formula is Taylor's theorem extended, and it is true for 
all values of h and k. When h and k are extremely small, the 
terms containing h^ , k^, and hk, may be omitted, and then dx 
may be written for k, and dy for k. This supposition will reduce 
the formula to 

Or du=^'idy+'^^dx. (7) 

dy dx 



tn DIFFERENTIAL CALCULUS. 

The expression ~dy represents the diflferential of the variable 
dy 

y in any function u, on the supposition that all else is constant, 

and it is called a partial diferenilal. 

Also — dx represents the partial differential of the function u 
dx 

in respect to x. 

In using this formula it is important to preserve the forms 

— dyt — dx, &c. otherwise we might confound these partial dif- 
cfy dx 

ferentials with the total differential du in the first member. 

Formula (7) is the same as (Rule 1,) (Art. 2,) and from that 

rule we infer at once that if m is a function of any number of 

variables, Siaf{x.y.s.t), then 

du=^d^'~u'^dy+^dz+^dL (8) 
dx ^ dy ^ dz dt ^ ' 

Art. 22.) Formulas (6), (7), and (8), should not be re- 
garded as equations of magnitude ; they are simply equivalent 
forms or symbols. 

Let us now examine formula (6). It can be put in this form, 

dxdy ^dy^ H \.^,^\dx^ ^ dx^dy ^ dxdy^ 

— ;[;A &o. &c. 
dy^ / 

If we conceive h and Jc to be extremely small, as we are at 
liberty to do, and then Write dx for A, and dy for Ic, the preceding 
formula becomes 

\dx ^dy " /^l.r<.dx' ^dxdy ' dy' ' )' 

1,2.3 Vda:" ^dx^dy "^dxdy* ' ^ dy^ ' / 



SEVERAL VARIABLE FUNCTIONS. 193 

Observe that the expression indicates that the function 

dxdi/ 

u is to be differentiated twice, once in respect to x, and once in 

respect to y. It is immaterial which differential is taken first, for 

d^u J d^u .J .. , 

, and , are identicaL 

dxdy dydx 

d^*^u 

The e:eneral expression , indicates that u must be dif- 

^ ^ dx'^dy'' 

forentiated {m-\-n) times, m times in respect to x, and n times 
in respect to y. 

Observe the last formula. Take the first parenthesis in the 
second member, 

du-, , du-, 

dx-\. dy, 
dx dy 

DijBferentiate each term twice, once in respect to x, and once in 
respect to y, and add the results together, and we shall have the 
term in the second parenthesis. 

Thus d('^^dx\=£^d=c-+-^l^dzdy. 

\dx / dx^ dxdy 

■j( duj \ d^u J 2 , d'^u , J 
d{ dy)=.^dy^-\-^—-dxdy. 
\dy / du^ dxdy 



By addition -^.( ^dx^ +^J^dxdy+-^dyA 

^ \.^\dx'' ^ dxdy ^ dy^ ^ ) 

Which is the second term of the formula taken as a whole. 

The differential of this again will give the next term, and thus 
we might go on indefinitely. 

Observe that the quantities in parenthesis tak« the form of 
an expanded hinomiul, and such in fact they are in a certain 
sense. 

(Art. 23.) Again let us inspect formula (6), for it is a very 
general formula including several rules and theorems. We may 
use it to develop the function of any two variables, however 
great the increments h and k. 



194 DIFFERENTIAL CALCULUS. 

If we suppose both x and h equal nothing, we have 

and this is Taylor's theorem. 

If we suppose x, h, and y, each equal nothing, and represent 



hy A, A^f Az, A^, &c. what ' 
Jiis supposition, we shall have 



du d^u d^u 



f(k)=A+A,k+A,. 



dy dy^ ' dy'- 



Ic^ 



, become under 



&c. 



1.2 • 1.2.3 • 

and this is Maclaurin's theorem. 

To illustrate these principles, we now give the following 



EXAMPLES. 



1. Expand x^y'^ by the differential formula (6) on the sup- 
position that X becomes {x-\-h) and y becomes {y-\-k)- 



Let 



u=:x^y'^, 



Take the first horizontal column in (6). 

The first term is x^y^ 

—-Jc 4x^y^k 

dy 

Sx^y^k'^ 



dy'' 1.2 

d^u k^ 
dy'' 1.2.3 

d'u k^ 



KM 



.x^k'^ 



dy^ 1.2.3.4 

Here the first column runs out, the last result x^k^ no longer 
contains y to admit of another differential in respect to that 
letter. 

Now we will run along the next horizontal column, taking the 
successive differentials in relation to y. 



SEVERAL VARIABLE FUNCTIONS. 



196 



du 

dx 



k 3a;2yU 



.^:'j^hk 

dxdy 

d^u hk^ 

dxdy^ 1.2 

d^u hk^ 

dxdy"" 1.2.3 

d^u hk^ 



dxdy'' 1.2.3.4 
Here the second horizontal column runs out. 



.12a:2y3^^ 
18a:2y2^F 
A'^x'^yhk^ 
.=3a;2M* 



(2) 



d^u h^ 
dx'' 1.2 



/ d^u h^ \k 

\dx^dy 2 /2 

VcfarVy^Y/Y 

/ d^u ^2\ ^3 

Vdz^dy^Y/ 273" 

/^^«z^\J^_ ==3xhn^ 

ydx^'dy^ 2 72.3.4 



. . . 3ary*^2 
.12a;y3A2^ 
18a;y^A2p 
. nxyh^k^ 



M3) 



Here the third horizontal column runs out. 
d^u h^ 



dx"" 1.2.3 



=:y^h^ 



( ^'^ -ii-V =42/3A3;5; 

\rfa;Vy 1.2.3/ j, r^\ 

2d term =6y^k^k^ 

3d term =4yh^k^ 

4th term =h^k'^ 

Here the process ends ; it is very easy when one is familiar 
with the forms. 

We will now do the same by common algebra. 



196 DIFFERENTIAL CALCULUS. 

x^y^-^4x^y^Jc-{-6x^y^k^-{-4x^yk^-{-x^k^ column (1) 
3x^y^h+l2x''y^hk-\-lQx''y^kk^-\-12x^ykk^+ 
Sx^hk*+3xy^k^+ &c. (2) 

These several columns are generally indicated by the corres- 
ponding columns in formula (6). 

We may use formula (7) to differentiate examples like the 
following, but the rules in Chap. I, are less formal and conse- 
quently more brief. 

2. Let «=-. 

y 

Then ^rf^=^, and ^y=^% 

dx y dy y 

Whence, by adding these results, we have 

y ydx — xdy 

u - 

3. Let u=. — . 

du-f^_ dx ^Vy- ^dy , 

dx Ji^y2 dy (i_y2ji 

dx xydy 

Whence du=~ 171+"^ X 

X X 

4. Let w=tan.~*_, or tan.«=_. 

y y 

duj _/ du \_dx duj _/ du \_ xdy 

dx Kcos.^u/ y * dy Vcos.^m/ y^ 

ydx — xdy 






y 

When - represents the tangent of an arc, the cosine of the 

y 

same arc must be ^ Whence du= ^ ^~^JL. 



PLANE CURVES, 



197 



CHAPTER VI. 

Application of the Difiercntial Calculus to discover 
some of the properties of Plane Curves. 



It is said by some, that the investigation of the properties of 
curves led to the consideration of flowing and vanishing quanti- 
ties, and from thence came fluxions, now called the differentia) 
calculus. 

Whether this be true or not, the following general problems 
will show the geometrical power of the calculus better than any 
thing thus far advanced. 

(Art. 24.) The theory we are now about to present to the 
reader is general, and therefore we shall refer to no particular 
curve until we apply the theory. We now propose to show ana- 
lytical expressions for tangents, sub-tangents, normals, and sub- 
normals, to curves in general. 

A tangent is a line drawn to touch the curve, and it is termi- 
nated by the point of contact and the ordinate. 

A normal is a line drawn perpendicular to the tangent from its 
point of contact, and it may be within or without the curve, 
according to the nature of the curve and the position of the 
ordinate. 

A line drawn from the point where the tangent meets the 
curve perpendicular to the ordinate, will divide the ordinate into 
two parts ; the part lying under the tangent is called the sub- 
tangent, and the part under the normal is called the sub-normal. 

According to these definitions, the reader will observe that in 
each of the adjoining figures, 1 and 2, MN\s a normal, FN is a 
stih-normxd, MT is a tangent, and PT a sub-tangent. 
Fig. 1. 






198 DIFFERENTIAL CALCULUS. 

Fig. 3. In every known curve some 

relation must be given be- 
tween AF and I'M, (fig. 3,) 
the co-ordinates of the curve. 
If we represent AP by x, 
and FMhj y, then y=f(^x). 
The similar triangles SPM, 
MQM\ give us the following 
proportion : 

SF : PM :: MQ : QM' . 

Now as we diminish PP' or h, the point M' becomes nearer 
and nearer to M, and the line M'MS revolving on the point Jfwiil 
bring aS^ nearer and nearer to T, and when M' comes to M^ then 
S will be at T, and the line MS become MT. 

But when h becomes extremely small, we call it dx, and then 

M'Q becomes dy, and corresponding to this the line MS becomes 

extremely near MT, so near that we may call it MT, then the 

preceding proportion becomes 

FT \ y \ \ dx \ dy 

"udx 
Whence PT=— — =suh-tanaeni. 

dy 

(Art. 25.) In the triangle TPMwe have 
{MTy = (TFy-\-(FMy. 

That is, {MTY=t^ 



dy- 



Or MT:=y l^^+\=the tangera. 

^' dy' 



The two triangles (Fig 1), TMF, MFN, are rectangular and 
similar. 

Whence TP : FM : : PM : FN 

That is, ^"^ '. y ::y : FN. 

dy 

Or PN=^-]l=z the sub-normal. 



PLANE CURVES. 199 

Also, TP \ TM \ \ MP : MN. 

That is, yA"". : y /^^l+l : : y : MK 

dy \ dy^ 

J- = (^F+0 ■■■''■■ (^^^'- 

Whence MN—^ Jdx^ J^-dy^= ike normal, 

dx 

It is obvious from the triangle MQM\ that the differential of 
an arc is Jdx'^-\-dy'^ , and calling the arc 5, we have 



ds=Jdx^-\^dy^. 
We will now collect these important expressions for future 
use, taken in the order of their development. 



sub-tan. =y^^. 
dy 

tangent =y^g-+l. 

sub-normal =^J^. 
dx 


(1) 

(2) 
(3) 


ormal = — Jdx"^ -\-dy^ 
dx 


(4) 



Differential of an arc =^ J dx^ -\-dy^ , 

APPLICATION OF THESE EXPRESSIONS. 

(Art. 26.) To apply these expressions to any curve, we 
must know the equation of the curve, otherwise we could not 
find dx and dy. 

1 . Find the sub-tangent, tangent, sub-normal and normal to the 
parabola, the equation being y2=2px. 

For the sub -tangent we must differentiate the equation, and 

reduce to the form ^— , and so on for the other lines. 
dy 

ydy=pdx. 

y dx / X 

p dy 



200 DIFFERENTIAL CALCULUS. 

p p dy 

Thus we discover that the sub-tangent of any parabola is (2a:), 
twice the abscissa, a result corresponding to (Prop. V.) page 252, 
Robinson's Geometry.* 

For the tangent we square (a), and i_=— - . 

p2 dy 



«' !^+-£+>' W^+-W£+>- 



p ^dy^ 



Whence the tangent =y. /_-}-l. 
^ P 



Sub -normal ==p. Normal =Jy^-\-p^. 

2. 5^Ae equation of ike ellipse, (the oriffin of the axes being the 
center,) is A2y2-|_B2x2=A2B2. 

What is the value of the sub-tangent? 

Ans, — -— ^. 
JS2 X 

What is the value of the sub-normal? 

B'x 



Ans. 



3. The equation of the circle is x ^ -J-y^ =R* , find the sub-tangent, 
tangent, sub-normal, and the normal. 

Ans. Sub -tangent = — ± — The minus sign indicates that 

X 

the sub-tansfent decreases as x increases. 



•&^ 



Ry 



Tangent =-^. Sub-normal =— a?. Normal =i2. 
x 

We shall apply these formulas to other curves, as occasion 
may require. 

The student will perceive that these results are here obtained 
far more easily than in analytical geometry, but we are indebted 
to analytical geometry for the primary equation of the curve. 

* It is important that the student should observe that this portion of the 
calculus is pure analytical geometry. 



MAXIMA AND MINIMA. 201 



SECTION II. 

CHAPTER I. 
maxima and minima. 

(Art. 27.) The diflPerential calculus embraces mathematical 
functions and geometrical magnitudes which admit of variation, 
whether increasing or decreasing in value. 

A differential of a quantity is an expression for a minute in- 
crease or decrease of the quantity. 

But when a quantity has increased to its maximum value, a 
further increase is impossible, and the expression of such an in- 
crease must therefore be zero. 

A decreasing quantity can of course have a differential, but 
when it has decreased to its smallest possible or minimum value, 
a further decrease is impossible, and the expression for it must 
therefore be zero. 

Hence the differential expressions for any function at its maxi- 
mum or minimum points must equal nothing. 

To geometrise this principle and make the idea visible, we pre- 
sent the following figure. 

Let AUD be a curve, and for the sake of 
perspicuity we will suppose it to be a circle. 
Let AC he one semi-diameter, and CD an- 
other, at right angles to it. 

Let us commence computation ftom the 
point A, and put AJB=x, and BE=^y. 

Now the magnitude of y will depend on that of jc, or in other 
words, y is a function of x, and it may be written 

y=/(*)- 

If X is increased by h, y must be increased by 1c, and by 
inspecting the figure we perceive that for equal increments of x 




202 DIFFERENTIAL CALCULUS. 

by h, the increments of y become less and less as JE approacbes 
D, and when i5jE' becomes CD, the increment A- becomes nothing. 
That is, the differential of j is zero when j itself becomes a max- 
imum. 

(Art. 28.) The diflferential of an increasing quantity is posi- 
tive before arriving at the maximum zero, and negative afterwards 
as we perceive by merely inspecting the figure, and this is a 
general principle. 

In like manner the differential of a decreasing quantity is 
minus before it attains its minimum point, it is zero at that 
point, and positive after passing that point. 

Hence, if the second differential of a function is minus, it in- 
dicates that the first differential corresponds to a maximum, and 
if plus it indicates a minimum. 

We will now work the example represented in the figure, 
which is this : 

What is the relation between the sine and versed sine of an are 
when the sine is a maximum^ 

Let i? represent the radius of the circle. 
Then by trigonometry 

y''=(2E—x)x. 

Qydy— 2Edx —^xdx. 

The first member contains dy as a factor ; and because y is to 
be a maximum, dy=^0 and makes the first member 0. 
Therefore {R — x)dxz=0. 

This equation will be verified either by making 

dx=0, or R — a;=0. 
That is, ar=0, or x=R, 

The first corresponds to the point A, where y ia sl minimum, 
and the last corresponds to CD where ^^ is a maximum. 

If we differentiate the equation (R — x)dx=^0, regarding dx as 
constant, we shall have — dx^=:0 for the second differential of 
the function, and it being minus, it indicates that the first differ- 
ential, or this factor of it, corresponds to a maximum. 



MAXIMA AND MINIMA. 203 

(Art. 29.) The foregoing illustrations are too plain and prac- 
tical to meet the entire approbation of some minds ; therefore, 
we give the following as more general and abstract. 

Let y=f(x)> and if y is a maximum it is greater than its 
corresponding value when we make x=x — h, or x=^x^h. Let 
y' correspond to {x — h), and y" correspond to [x-\-h). 

Then, by Taylor's theorem, we have 

^ ^ dx ^dx^ 1.2 dx'' 1.2.3^ ^ ' 

And y"—7/=^k-l-—^ -\--^ , &c. (2) 

^ '^ dx ~dx^ 1.2 ^ dx^ 1.2.3 ^ ^ 

Divide (1) and (2) by A, and 

yW^..^ , ^>J__ &c. (3) 

k dx^dx"- 2 ^ ^ 

fziy^ciy.d^k, ^^^ .^. 
h dx^dx^ 2^ ^ ' 

Now h can be taken so small that ^ _Z j in (3) and (4) will 

be greater than all the following terms, and we make this sup- 
position. 

Then if y is greater than both y' and y" the sign of the first 
members of (3) and (4) are both minus. Therefore the sign of 
the second members must be loth essentially minus. 

But this cannot be unless — =0, and therefore this condition 
dx 
must exist. 

Now suppose that y be less than either y' or y", then the sign 
of the first members of (3) and (4) must both be plus, and 
hence the sign of the second members must both be plus, but 

this cannot be unless -K=0. 
dx 

Therefore when y, a function of x, is a maximum or a mini- 
mum, -^=0, or dy=0, 
dx 

MISCELLANEOUS PROBLEMS IN MAXIMA AND MINIMA. 

From the nature of the question or problem, find a general 



204 DIFFERENTIAL CALCULUS. 

algebraical expression for tlie quantity that is to be a maximum 
or minimum, and pronounce it sucb. Then its differential can 
be put equal to 0, and a solution of this equation will answer 
the question proposed. We must find as many independent 
equations as the problem contains variables. 

1. Divide a line or any given numerical quantity (a) into two such 
parts that their product will be a maximum. 

Let x= one part, then a — a;= the other part. 
The problem demands that {ax — x^) shall be a maximum, and 
this is the same as to demand that its differential shall be =0. 
Whence {a — ^x)dx=0, or x=^\a, Ans. 

2. Divide a given quantity (a) into two suck parts thai the 
'square of one part multiplied by the other part shall be a maximum. 

Ans, The part to be squared is (Ja). 

3. Divide the number 80 into such parts, (x) and (y), that 
2x^-|-xy+3y^ may be a maximum or minimum. 

Ans. x=.50, 2/=30, for a minimum. 
For a maximum 3/= 80, and x=:0. 
Here we have two equations 

^+y=80, (1) 

And 2x^ -\-xy-\-3y'^ = maximum or minimum. (2) 

The differential of the first equation is zero, because it is con- 
stant; the second is zero, because it is a minimum. 

4. Mnd the greatest rectangle that can be inscribed in a given 
triangle. 

JAns. The altitude of the rectangle is one half the 
altitude of the triangle. 

Let ABC he the triangle. 
AB=b, DC=a. Take CI=x. 
Then x '. EF \ \ a \ b. 



G^ D b: B 



EF^—. ID=a~x=^FH. 
a 

bXf \ 

— {a — a;)=:maximum. 

a 



MAXIMA AND MINIMA. 206 

The differential of this expression will contain /_ j as a com- 
mon factor. The product made by it and another factor must 
equal zero. Therefore the other factor alone must equal zero, 

for (-)> known constant values, cannot equal 0. 

Hence we may have x(a — a;)=maximum. 

That is, constant factors to the whole membery expressing a maxi- 
mum or a minimum, may he omitted before differentiation. 

5. Find the greatest rectangle that can be inscribed in the 
quadrant of a given circle ^ 

Ans. The rectangle is a square. 

Remark.— In these philosophical mathematical problems the operator 
Bhould not consider himself restricted to any mere rules ; he is at liberty to 
apply general principles in their widest sense ; the following example is an 
illustration of this remark, — observe its solution. 

6. If two given circles cut each other, find the greatest line that 
can be drawn in them passing through either point of intersection* 

Ans. The line is parallel to that joining the centers. 

Let BD=^x, and 
put R to represent 
the radius of the cir- 
cle. 

Let AC=7/, and 
put r to represent the 
radius of that circle. 




Then OD=JE^—x\ and CO=Jr^—y\ 
Because w (7= CO and OD=J)n, 



2jM^'-x^+2jr^—7/^=msLX. 



Or 7i23_^2_|_^^2_2^2^jnax. 

Let AB, the distance between the two centers be represented 

» This is problem 9 page 207, Mathematical Operations. 
14 



206 DIFFERENTIAL CALCULUS. 

by a, then CH=ia, and DH=x — y, and the right angled triangle 
CDHgi\QS 



In every equation the d'fferential of one memher taken as a whole 
«^ equal to the differential of the other. 

But the first term of this last equation is a maximum, there- 
fore its differential is 0, and the differential of the second mem- 
ber is 0, because it is invariable. Therefore 

d.{x—yY=zO. 
Or ^{x—y){dx—dy)=0. 

Whence x — y=^0, or {dx — dy)=0. 

From either of these x=y, which shows that the line through 
must be drawn parallel to AB. 

7. From two given points on the same side of a line given in po- 
sition, draw two lines to meet in the line given in position, whose sum 
shall he less the sum of any other two lines drawn from the same 
points to the same line. 

iAns. The two lines make equal angles with the 
( line given in position. 

Let A and B be the two points and ffO the 
line given in position. 

From A and £ drop the two perpendiculars 
B and AIT, and these lines are given because 
the position of the two points are given ; and 
for the same reason Off is given. 

Make OF=x, and JSff=y, B 0=a. AFI=h, and ffO=c. 




Then Ja^+x''=BE, and Jb^+y^=AE. 

Now the problem requires that 

J a 2 -\-x^ + Jb 2 -\-y^ =minimum. 
Whence _^_+_,^^=0. (1) 

But x-\-y=c. Hence dx-\-dy—0. (2) 



MAXIMA AND MINIMA. 207 

From (2) we have dx= — dy, and this value of dx put in (1), 
and that equation reduced, we find 

^ -. y 



By inspecting the figure we find 

OE__EE 
EB EA' 
That is, EB : OE : : EA : EH, 

sho'^ring that the two triangles jB^Oand AEH&re equi-angular, 
and the angle BEO= to the angle AEH. 

The angle AEIIis equal to the vertical angle OED, and BO 
produced will make 0D= OB. 

Hence to find the point required, produce B 0, making 0D=^ 
OB, and join AD cutting OH in E. Join BE, and we have 
AE and EB, the two lines required. 

Let OH he a plane mirror, B an object, and A the eye of an 
observer, the object B would be seen below the mirror at D. 
Hence, rays of light reflected . from a surface take the shortest 
possible distance in passing to and from the reflecting surface. 

This problem shows us the truth of the definition that the cal- 
culus is a branch of analytical geometry. 



8. Required the greatest possible rectangle that can be inscribed 
in a given parabola. 

\Ans. The altitude of the rectangle is | the 
altitude of the parabola. 



I 



9. Required the same in a given semi-ellij^se. 

Let 2/= the altitude of the rectangle, and B half the shorter 

7? 

axis of the ellipse, then we shall find that y=. _., the result. 

10. Required the maximum cone thai can be inscribed in a given 
sphere. 

j Ans. The altitude of the cone is |- the radius 
( of the sphere. 



«08 DIFFERENTIAL CALCULUS. 

1 1 . Required the relation between the diameter and altitude of 
a cylindrical cup to hold a given quantity (a) of water, and to con" 
tain the least possible surface. 

Ans. The radius = the altitude. 

1 2. Required the maximum parabola that can be cut from a given 
right com, 

J Ans. The axis of the parabola is equal to f the 
side of the cone. 

13. Required the maximum cylinder that can be cut from a given 
right com, 

iAns. The altitude of the cylinder must be ^ the 
altitude of the cone. 

14. On a horizontal plane stands a tower 60 feet high, and <m 
the tower stands a spire 20 feet high; hoto far from the foot of the 
tower will the spire appear under the greatest possible angle, and 
what will that angle be? 

Let x= the distance from the foot of the tower. 
Put a=60 feet, 5=20. 



Then ar=7(a+5)a==4073=69.28 feet, Ans. 

This example is a very simple one, but we solve it to explain 
one important expedient that may often be resorted to in working 
questions in maxima and minima. 

Let a;= the distance on the plane from the tower, and put A 
to represent the whole angle at that point between the foot of 
the tower and the top of the spire. 

Put B to represent the angle from the foot of the tower to its 
top. Then {A — B) is the angle to be a maximum. 

By trigonometry, 

1 : tan.^ : : a; : c+J, tan.-4=— i-. (1) 

X 

1 : tan.J5 ii x x a, tan.i5=-. (2) 

X 



MAXIMA AND MINIMA. 209 

b 

But taD.M-~^)= ^^"-^-^-^"-g .^ ^ 

^ 1+tan.^ tan. i^ i i («+^)a 

Or tan.(^-^)=^,,^^. (3) 

The arc (A — B) is to be a maximum, not the tan.(^ — B). 

But if we differentiate the first member of the equation, it 
will contain d(A — B) as a factor, and as this factor must be 0, 
the product of all the factors will be 0, and therefore the differ- 
ential of the second member must be 0. 

That is bdx{x^+ {a+b)a) —^bx^dx 

(x^+ (a+b)ar 

Whence x=J(a-\-b)a. 

To find the magnitude of the angle (A — B), we substitute 
this value of x in equation (3), 

Tan (A-B)J^^^^=—J=^=-^-^-^' 
^^•^ f 2(a+6}a 2V(a+6)a 2.40V3~473 

This result corresponds to radius unity; multiply it by i?, the 

radius of our tables, (as follows:) 

R log 10.000000 

4^3 log 0.840620 

Tan.(^— ^)=8° 12' 47" 9.159380 

15. An architect was required to give the relative length, breadth^ 
and hight of a rectangular building , to contain a given cubical space 
(a) to be enclosed, sides, top and bottom, by the least possible surface. 

Ans. The building must be a cube. 

16. Divide a given numher (a) into three parts so that their con- 
tinual product may be a maximum. 

Ans. The parts must be equal. 

17. Find the minimum value of y in the equation y=x*. 

Ans. y^^=Y\. 
N. B. Take log. of each member, then differentiate and place 
dy—0, and we shall find log.a:-|-l=0. 



210 DIFFERENTIAL CALCULUS. 

18. Two roads, one exactly north and south, the other exactly 
east and west, intersect each other. One traveler ten 7niles north of 
the intersection^ starts and travels south at the rate of four 7niles per 
hour. Another traveler six miles west of the intersection, starts ai 
the same moment and travels east at the rate of three miles per hour. 
Hovj long after starting will they he at the minimum distance from 
each other, and what will that distance he^ and what will he the locality 
of each? 

Ans. The time will be 2//-0 hours. The one traveling south 
will be yVo" of a mile north of the intersection ; the one 
going east will be yVo- of a mile east of the intersection, 
and their distance asunder will be ly\ miles. 

19. JVbw suppose two roads to intersect as before, and one trav- 
eler to start from a 7niles north of the intersection and travel south 
at the rate of m miles per hour, and the other traveler at the same 
time to start frcyiih b miles west of the intersection and travel east at 
the rate of n miles per hour, what time must elapse hefore they arrive 
ai a minimum distance, and what will that distance he? 

Ans. Let t=^ the time. Then t^-^A"^ 

m^'-^-n^ 

Distance =J{a — mty-\-{b—nt)^. 

Any number of numerical examples can be formed from this 

one by giving different values to a, h, m, and n. 

20. The difference of arc between the sun's right ascension and 
Us longitude gives rise to one part of the equation of time. What is 
the sun's right ascension when this part of the equation is a maxi- 
mum, and what is the maximum value? 

*Ans. Sun's Long. 46° 14' 10" 
R. A. 43° 45' 60" 



Difi*. 2° 28' 20"=9m 54.6 *. 

21 . What must be the inclincdion of the roof of a building to make 

the water run off in the least possible time? 

Am. 45°. 

« Examples 20, 21, and 22, are solved in the author's Mathematical Ope- 
rations. 




MAXIMA AND MINIMA. 211 

22. Within a triangle is a given point P, the distance to the near- 
est angle A is given, and the line AP divides the angle A into two 
angles m and n, of which m is greater than n. 

It is required to find the line EF dra-ivn through the point P, so 
that the triangle AEF shall be the least possible. 

Let AF=a, AF=x, AB=y. The 
angle FAF=m, FAIJ=7i. 

The area of the A AFF=axsm.m. 
The area of the A AFE=ay sin.w- 
By the conditions, 

ax ^vQ..m-\-ay &m.n= minimum. 
Also, by the conditions, 
rrysin.(?7i-|-?2)= minimum. 
\Ans. The line ^i^must be drawn so as to make 
( the triangle APE^ to the triangle AFF. 

23. What is the altitude of the maximum cylinder which can be 
inscribed in a given paraboloid? 

Ans. Half the axis of the paraboloid. 

24. Conceive an ellipse to revolve on its longer axis, thus forming 

an ellipseoid. Find the maximum cylinder which can be cut from 

this ellipseoid? 

4 Ti 
Ans. The diameter of the cylinder is 

Its solidity is ![?:iilil£:^. 

A represents the major semi-axis of the ellipse, and B the 
minor semi-axis. 

25. Find the least triangle that can be made to enclose the qua- 
drant of a given circle. 

Ans. The point of contact is at the middle of the arc. 

26. There is a perpendicular chimney; width of cavity h inches, 
kight of the jamb above the floor a inches. Mequired the longest in- 
flexible pole that can he put up the cavity. 



Ans. Ma' + ^'^y 



Av 






^/ 




Yc 




^^. 



212 DIFFERENTIAL CALCULUS. 

27. It is required to determine the size of a hall which heing let 
fall into a conical glass full of water, shall expel the most water pas- 

siblefrom the glass; its depth being 6, and its diameter 5 inches. 

Let ^j5 (7 represent the conic section of the 
glass, and DJIJS the ball, touching the sides in 
the points J) and B, the center of the ball being 
at some point i^in the axis GC oi the cone. 

Let FD=FE=x, the radius of the sphere, 
then find an expression for the magnitude of 
the segment of the sphere immersed in the 
water, and this segment must be a maximum. 

Ans. a;=2^^ inches. 
An equation may have more than one maximum or minimum, 
according to the degree of the equation, as the following exam- 
ple will show. 

Let x^ — ^x^-\-9.2x^ — 24a;-}-12= a maxima or minima. 

Then Ax^~Ux''-\-Ux—M=^0. 

Or a;3— 6a?2+lla;— 6=0. 

Whence ir=l, or 2, or 3. Substituting 1 in the equation, and 
we have 3, a minimum. Substituting 2, and the value of the 
equation is 4, a maximum. Again, substituting 3, and the equa- 
tion produces 3, a minimum. 

28. Required the least triangle that can he drawn ahout a given 
parabola. 

iAns. The sub -tangent on the axis is two-thirds 
of the given axis. 

29. Required the same ahout a given semi-ellipse. 

N. B. In solving this, we use the sub -tangent taken from 

A^v^ 
(Art. 26,) which is — — ^, but we change its sign, for the 

signs in geometry refer only to direction, and not to numerical 
values. 

The zero point being the center of the ellipse, if we commence 
ill the left or longer axis produced, and reckon towards the right, 



MAXIMA AND MINIMA. 



213 



our distances will be plus all along the base of the triangle, be> 
cause that direction is plus along horizontal lines as the upward 
direction is plus on perpendicular lines. 

If we put z to represent the altitude of the triangle, x and y 
being co-ordinates of the tangent point, we shall have 






y ' 






X-x : z= -. 

y 



In conclusion we shall find 2/= — -, and :c= — _. 

If we compare examples 29 and 9, we shall find that the inte- 
rior maximum triano-le and the exterior minimum triangle of the 
ellipse meet the curve in the same point. The same is true in 
respect to the circle and the parabola. 

30. It is required to cut the greatest possible ellipse from a given 
right cow. 

Let AH=^a, the base of the cone, and V, 
the vertex of the cone, be h distance above 
the base. 

Let AB be the greater axis of the ellipse. 
Let fall BP, the perpendicular, on the base, 
and take G, the middle point between B and 
II, and pass a horizontal plane through the 
cone parallel to the base of the cone. 

This plane will cut the plane of the ellipse at the center C, 
and CD will be the minor axis. 

As C is the middle point between A and J5, and G the middle 
point between B and II, it follows that (7(7 is the half of AH. 

Fui HP=x, PB=^y. Imagine a perpendicular from Fto 
AH, which is b. Then we have the following proportion : 




X : 2y : : a : b. Or x=- 



(1) 



Also, AP^+PB^=ABK That is, (2a—xy-}-4y^==ABK 



The greater axis of the ellipse =V^'"^ — 'iax-\-x^ ■•\-4j/'-^ . (2) 



214 DIFFERENTIAL CALCULUS. 

We now require the value of GK. The perpendicular from 
Fto AH\^ b. From Fto GKis b — y. Therefore we have 

b—y : GK :-, b : 9,a. GK=<^a—?f:t 
^ b 

From this take GO (a), and we have KC=a——l. 
^ ^ b 



But KC, CG= CD'' . That is, ^a^ _^^ ^ cD, minor axis. 

But the product of the major and minor axes of an ellipse de- 
termines its area. When that product is greater, the area is 
proportionally greater, and when less, less. 



Therefore (J^cl^ — 4ax-\-x^ -\~4y^ ) Aa^ — — ^^=max. 

Or (4a2__4^^_|_<^2 _|_4y 2 ^ / 1_ ^y \ ^j^aximum. 

Taking the value of x from (1), and substituting it in the 
above, we have 

(4a^-^+ii:^+4y^) (*-=5?^)=niaxiinum. 
Whence (z:^Uy+^dy+^dy)(t:ll) 

Dividing each side by — -, and we have 
b 

(=4^1+4«!,+4y)(5_2y)=4a^-i^+4^^!+4y'. 

~ b ~ "^ b b' b ~ b' 

b- 

Dividing by — 4 produces 



MAXIMA AND MINIMA. 216 

^ V36-^+3aV"^ 3b^-\-^a^ 

Remarks.* — The shape of a cone depends on the relative 
values of a and h, b must be greater than a, or y will be imagi- 
nary in the above equation, showing that the oblique elliptic sur- 
face will not be greater than the horizontal base of the cone. 
And to render a maximum ellipse possible, the relative values 
of a and b must be so taken in the last equation that y will have 
a positive value. 

To be sure of obtaining real values of y, the square of half the 
coefficient of j must be numerically greater than the second member^ 
or at leo^t equal to it. 

That is, we must have -i L — i = , at least. 

4(3^2 _|_3^2j2 352_|,3^2' 

Or I^^^=8a^ 

362-1-3^2 

Or 26a''+10a232+5*=24a252 4-24a^ 

Or a4_|_54==i4a252_ 

Add 2a^6^ to each side, to make complete squares. 

Then a^+2oJ'h^-\-b^ = \Qa''b''. 

Square root a^-|-^^=4a5. 

If we put b^=ma, this last equation reduces to 1+^^=4^, 
and this resolved, gives m=2=b^3=3.732. 

This shows that b must be greater than (3.732) times a, other- 
wise y will be imaginary in equation (3), and the circular base 
will be greater than any ellipse that can be cut from the cone, and 
in that case no maximum ellipse luill be possible. 

We may therefore take b of any value greater than (3.732), 
we will then assume i=4a, and this reduces (3) to 
51a2y2_84a3^^_2a2 je^2^ 

Whence y= 1.048a nearly, or |a nearly. 

•These remarks show that this whole subject is one of analytical geometry. 



816 DIFFERENTIAL CALCULUS. 

It may be that if a plane be passed through to the opposite 
extremity of the base of the cone from each of these points, the 
elliptic surfaces will be the same, and greater than any other 
above, below, or between, then and there are two maximums. 

31 . It is required to find that fraction which exceeds its square by 
the greatest possible quantity. 

Ans. +I-. 

32. It is required to find that fraction which exceeds its cube by 
the greatest possible quantity. 

Ans. +-i^. 
V3 



CHAPTER II. 

On the signification of BiflTerential Coefficients as 
applicable to Curves. 

(Art. 30.) In analytical geometry a curve is traced by con- 
necting different points found by an equation — the equation to 
the particular curve in question. 

The nearer a curve is to a right line, the less will be the value 
of the second and third differential coefficients — and when the 
curve becomes a right line, the first differential coefficient is 
constant, and the second, third, and all the following differential 
coefficients are zero. 

For example, the equation of a straight line is 

yz^ax-^-b. 

Whence -^^a, and _'^'^=0, &c. &c. 

dx dx^ 

The first of these differential equations is the differential equa^ 
tion of a right line, and let the reader observe that it is the trig- 
onometrical tangent of the angle which the line makes with the 
abscissas. 



THEORY OF CURVES. 217 

But before we proceed with our theoretical investigations we 
will draAv out and arrange the following differential equations. 
In analytical geometry we found 

1. x^-\-y^ =2i^, to be the equation of the circle. 
From which we find 

dx y 
for the differential equation of the circle, 

2. From {A'^y'^^-B^x^ —A^B"^), the equation of the ellipse, 
we derive 

«^___ B^x 
dx A^y 

tJie differential equation of the ellipse* 

3. From {y^==2px), the equation of the parabola, we 
derive 

dx y 
for the differential equation of the parabola, 

4. From (A^y''--B'^x'+A^B^=0), the equation of the 
hyperbola, we derive 

dy__ B^x 
dx A^y* 
for the differential equation of the hyperbola. 

5. From (xy=M), the equation of the hyperbola referred 
to its center and asymptotes, we derive 

dx x 
for the corresponding differential equation of the hyperbola. 

Thus, every curve that can be indicated by an equation has 
its corresponding differential equation. 

dy 
When ^ is applied to a right line, it is the trigonometrical 



218 



DIFFERENTIAL CALCULUS. 



tangent of the angle included between the line and the axis of 
abscissas; therefore we can use its equal for that quantity in ana- 
lytical geometry. 

Conoeive a right line touching an ellipse in a single point, the 
co-ordinates of which are x', y', then as above 

~dx' A^y'' 

But a line passing through a given point is represented by the 
equation y — y'^=ia{x — x'), 

as is well known by all readers of analytical geometry. 



But 



Therefore 



dy'^ ^_B^x 
dx A^y 



y—y'-' 



B^x 



-{x—x'). 



Reduced 



A^yy'+B^xx=A^B'', 
Which is the same equation for the tangent of the ellipse as 
may be found in analytical geometry, x and y being the general 
co-ordinates of the line, and x' y\ the co-ordinate of the par- 
ticular point touching the ellipse. 

Thus we may find the equations for the tangent lines to all 
known curves. 

Examples like this serve to 
impress upon the minds of learn- 
ers the connection between this 
analysis and analytical geometry. 

(Art. .31.) We shall now at- 
tempt to show the analytical ex- 
pressions for the deviation of 
curves from a right line in the 
vicinity of a given point, and 
also what uses can be made of 
such expressions. 

Let AP^x, and PM—y. 
Then 3/=/(^)- 




THEORY OF CURVES. 219 

Put PF^K and FP"=2k. 

Then ^'^'=/(-+/0=y+(|>+^^^^^ 

And P"M"=^f{x+^h)= y+^|^2A+^g)J^+ <fec. 

Whence P'M'--y=OM'=(il\h+( '^l1\]!l.+ ^q. (1) 

\dx/ \dx^/l,2 

And P"M"—y=J}^M"=ffl)2h+f—^Y-^+ &c. (2) 

\dx/ \dx^/l.2 

Because MN=2M0, NS=20M\ 

That is, NS=2 OJf' =-^2^+^!^-^+ &c. (3) 
c^a; ^dx^ 1.2^ ^ ^ 

Frona (2) subtract (3) and we shall have 

NM"—NS=—lh^ + &c. (4) 

dx^ 

Now if A be taken sufficiently small, the sign of the first term 
will be the sign of the sum of all the terras. 

The first member of (4) is obviously ijositive, and the curve 
being above the axis of X, all the ordinates are positive. 

Hence, when a curve is convex towards the axis of abscissas and 
the ordinates positive, or the curve above the axis of X, the ordinate 
and second differential coefficients vjill have the plus sign. 

Let us now examine the curve below the axis of X, which is 
also convex towards the axis of abscissas. 

Here AP=x, Pm^~y, PP'=zh, PP^^Ih. 

Whence P'm'=— /(y+A)=— y— /f^^^V" (^--^V— — &c. 

\dx/ \dx^/1.2 



220 DIFFERENTIAL CALCULUS. 

\dxj \dx^J\,2 ^ 

As ns is double nn\ we have 

\dxj\ \dx^A'2 ^ ^ 

' Subtracting (3) from (2), and we obtain 

sm"=nm" — ns= — ( — ±]k^ — &c. 

\dx^y 

It is obvious that sm" is the deviation of the curve from the 
right line, and it is minus downward, as the ordinates are. 

Hence, when the curve is convex toward the axis of abscissas, and 
the ordinates minus, or Hie curve below the axis of X, the ordinates 
and the second differential coefficient will have the minus sign. 

More generally, let the curve be above or below the axis of 
X, and convex towards that axis. 

Then the ordinates and the second differential coefficient will have 
the same sign. 

(Art. 32. ) Now let us determine what the result must be 
when the curve is concave towards the axis of the abscissas. 

Let AP=x as before, and PM=y, FP'^h, and PP"=2h. 

PM=y=f{xy 

And P"Jlf"=/(^+2A)=2,+(|)2A+(g)g+ &c. 

But 2y5=2iWV"=/^^V+/^-— V-+ <fcc- (2) 
\dx/ \dx'^J\.2 



\ 



THEORY OF CURVES. 



221 



Subtracting (3) from (2), and 
we obtain 

Now h may be taken so small as 
to make the first term in the e 3ond 
member numerically greater than 
all the terms that followed, and as 
we have its square, the sign cannot 
be affe<;ted by the essential sign of 
h, hence the second member of the 
equation is negative — which is also 
shown by the figure, the point M" 
in the curve is below the line MS. 

But in this case the ordinates are 
positive, hence the ordinates and the 
second differential coefficient have 
diflferent signs. On a like exami- 
nation of the points of the curve 
below the axis of Xwe shall find the same result. 




Hence, if a curve is concave towards the axis of abscissas the or- 
dinates and second differential coefficient will have contrary signs. 

For an example to apply this theory, let it be required to 
determine whether the parabola is convex or concave towards 
the axis of abscissas. 



dx y 



c?^_ pdy ^ 

dx^ y^dx 



The last equation gives a clear response, for the quantity — tL. 

is obviously negative when y is positive, and positive when y is 
negative, therefore the curve is concave towards the axis of ab- 
scissas. 

Thus we might determine the position of the concavity of any 
other curve. 

(Art. 33.) A point of infection is a point at which a curve 
16 



222 DIFFERENTIAL CALCULUS. 

changes from convex to concave, or from concave to convex, 
towards the same line. 

When a curve is convex towards the axis of abscissas, the 
ordinates and second differential coefficient have the same sign, 
and when concave towards the same axis, those two quantities 
have different signs. Therefore if a curve changes its position 
of convexity, the second differential coefficient must change sign 
at the point of inflection. 

But when a quantity changes sign it must pass through zero 
or infinity; hence, 

d^y ^ d^y 

— ^=0, or — ^=00 

dx"" dx^ 

will give the abscissas of the point of inflection. 

To find a point of inflection we will therefore put the second 
differential coefficient equal to or infinity, and determine the 
value of X, which value we will increase and diminish by a small 
quantity h, and if we find contrary signs for these new values 
of X, we must conclude that here is in fact a point of inflection. 

The following general equations represent an interesting class 
of curves which serve to illustrate this theory. 

y=l+{x-ar. (1) 

Whence ^=m(a;— a)"°-«. (2) 

^=m(m-\){x-a)-K (3) 

1st. If we assume that~=0, it follows that a;=a, and this 
dx 

value put in (1) will give y=5. 

But to assume that — =0, is the same as 
dx 

to assume that the tangent line through the 

point M in the curve is parallel to the axis 

of X, as represented in the figure. 

If m represents an entire and even number, 

then {m — 2) will be even, and all values of Xy except jr=a, will 




THEORY OF CURVES. 223 

give 2/and^ — | positive, for suppose x=a±:h, then (x — a)=dzh, 
and substituting this in (3) we obtain 

g=m(m_l)(±;.)"-^ 

Here as (m — 2) is even, the power of k will be positive, 
whichever sign we give to h^ and as m is even, the whole product 
will be positive, hence this curve is convex towards the axis 
of X. (Art. 31.) 

2d. Now let m he an entire and odd number. 

Then, as before, when x=.a, — =0, and the second difFeren- 

dx 

tial will also equal 0. 

But since (ni — 2) must be odd, every value of x less than a 
will make the second differential coefficient negative, and every 
value of X greater than a will make it posi- 
tive: hence, for all values of x less than a, 
which give y positive, the curve is con- 
cave towards the axis of X, and for all 
values of x greater than a, it is convex, as 
in the figure adjoining. 

Therefore at the point M, the co-ordinates of which are a;=a, 
y=5, the curve changes from being concave, and becomes con- 
vex towards the axis of X. 

If the last term of equation (1) be negative, that is, if 
y=-h—{x-^y, 
the reverse position will correspond with the curve, as in the next 
figure. 

At the point M whose co-ordinates are 
x=-a, y=-hy there is a change of conxexity 
to concavity towards the axis of X, 

Such points are by some called singutar 
points, — by others they are denominated 
points of inflection. 

In both cases the tangent line at the point of inflection is 
parallel to the axis of X, and it also cuts the curve. 





224 



DIFFERENTIAL CALCULUS. 



3d. Let m he a fraction, the numerator and denominator of which 
are odd, as |. 



Then 



d^y_ 



6 



dx 



dx^ 



And if we now take a;=a, we shall have y=5, 

■J'= infinity, and — 1= infinity, &c. 



dx 



dx"" 




Now if we suppose x less than a, — ^ will be positive, and if 
greater than a, negative. 

Hence for all values of x less than a, which 
give y positive, the curve will be convex, and 
for all values of x greater than a, it will be con- 
cave towards the axis of X, as shown in this 
figure. 

But if the binomial term be negative, that 
is, if we have 

3 

y=:zh—{x—ay, 

the second diflferential coefficient will be positive, and the re- 
verse will be the case as represented in the next figure. 

The point M, whose co-ordinates are 
ar=a and y=-h, in both cases is a point of 
inflection at which the tangent line is per- 
pendicular to the axis of X. Whence we 
may say, a point of inflection is one at which 
as the abscissa increases, a curve changes from 
concave to convex, or the reverse, towards any 
right line not passing through the point. 

4th. Let m he a fraction with an even numerator, as |, then 

2 

dy_ 2 d^y_^ 2 




dx 



S(a;--a)3 



dx' 



9(a;— a)3 



I 



THEORY OF CURVES. 226 

Tf x=ia, v=h. — ^= infinity, and — ^= infinity. 
dx dx^ 

If X is less than a, — will be negative, and if x is greater than 
dy 

a, it will be positive. Hence, at the point whose co-ordinates 

are x=a and y=^h, ^ must change its sign from minus to plus, 
dx 

which change indicates a minimum ordinate. 

If the sign before {x—a) be negative, the reverse will be the 
case, and there will be a change from jplus to minus, indicating 
a maximum ordinate. 

In the first case the second dijfferential 
coefficient is negative for all values of a?, 
and the ordinate positive, the curve is there- 
fore concave towards the axis of Xy as rep- 
resented in the adjoining figure. 

In the second case, that is, 

2 

the second 'differential coefficient is always positive for all 
values of x, (except a;=a). Then for all positive values of y, 
the curve will be convex, and for all negative 
values of y, concave towards the axis of 
X, as this last figure illustrates. 

The tangent at the point M, in both 
cases, is perpendicular to the axis of X. 

The point Mis singular and is called a 
CUSP of the first order. 

It is a point at which apparently two curves unite, but it is 
really the same curve, as one equation represents any point 
either branch. 





iji 



5th. Let m he a fraction of an even denominate , as f , 

Since the denominator of the fraction denotes square root, th© 

3. 

double sign must be placed before {x—aY, and we have 
yz=b-\-{x — a) 2. 



2*26 



DIFFERENTIAL CALCULUS. 



dx 



==bi(a;— «)2. 



d'y. 



dx"- 



Ajx — a 



x=ia ffives y=5, — =0, and — ^= infinity. 
^ ^ dx dx^ ^ 

If x is taken less than a, y will be imaginary, showing that 
no ordinate from a point nearer to the origin than a, can meet 

fin J d^ IJ 

the curve. If x be taken greater than a. ~^ and 



will be 




c?a; dx' 

real quantities with the double sign dc, show- 
ing two branches of the curve, as the figure 
represents. 

The point J!f is a cusp^ and the tangent at 
the point M is parallel to the axis of X, 



(Art. 34.) To draw out a little more light on the theory of 
curves, which is considered by mathematicians as one of the 
most beautiful features of the calculus, we will take the equation 

y=x^±x^. 



(1) 



dx 



3. 

^Xzti%X^. 



(2) u 






(3) 



When a:=0, y^=0, hence the curve will pass through the origin. 
If X be negative, y will be imaginary ^ because the equations 
would then demand the square root of negative quantities which 
have no existence, hence no part of the curve is on the negative 
side of the axis of Y. We also perceive 
that for every positive value of x there are 
two real values of y, both of which are 

positive as long as x^ is greater than x^\ 
after which one is positive and the other 
negative. 

When aj=0, ^=0. Also, the first difi"erential is 0, when 
dx 




2:»±|a:2=0. 



THEORY OF CURVES. 227 

Whence x=0, or a;=^f, indicating that the axis of X\s tan- 
gent to the curve at the origin, and the tangent to the lower 
branch must be parallel to that axis at a point whose abscissa 
is H- 

The first value of -irj belongs to the upper branch of the 

curve, and it is always positive. The second value is also posi- 
tive as long as 2 is greater than '-^ Jx. Hence, the point that 
corresponds to 

must be a point of inflection whose abscissa is 2V5- 

Hence, the preceding figure represents this curve, and its 
origin is a cusjp of the second order. 

(Art. 35.) In analytical geometry (page 118) we delineated 
the curve corresponding to the equation 

y=a;3— 18a;+12, 

and there gave the maximum and minimum points corresponding 
to y. But the determination of those points of course depended 
on the calculus, which the reader was not then supposed to un- 
derstand, and we now notice the fact to show that the subject 
of curves requires the calculus to be complete. 

By taking the values of -^, and — ^, in connection with their 
dx dx^ 

signs, we can determine the concavity of the curve at any as- 
sumed point. 

Thus ^2^=3^^—13. (1) ^-!^=6:r. (2) 

dx dx"^ 

If we put 3.'i;2_13=0, we shall find .'r= ±2.08 14, showing 
two points at which the tangent is parallel to the axis of X. If 
in the equation 

|^=3a,-^-13, (3) 

dx 

we assume a:=0 we shall have -^= — 13, showing that a tan- 

dx 



228 DIFFERENTIAL CALCULUS. 

gent to the curve at the point where it cuts the axis of Y is 
—13, or 94° 24' with the axis of X. 
If in (3) we make a;=4, we shall have 

^=35, 
dx 

showing that at that point the natural tangent with the axis of 
Xis 35 to radius unity, or 88° 21' 49". 

If we make x negative in (2) while y is positive, the curve 
will be concave towards the axis of X. (Art. 32.) 

If we make x positive in (2) while y is positive, the curve at 
the corresponding point will be convex towards the axis of X, as 
is already shown by the construction of the figure. 

(Art. 36.) Curves are sometimes 
accompanied by insulated pointSt as 
the following equation will illustrate, 
a^if=x^—.hx''. (1) 




Or yz=z± ?J^—b , (2) 

a 

In either (1) or (2), if we make 
2;=0, we shall have y=:0, therefore 
the origin is a 'point in the curve. 
But on inspecting (2) it is obvious that y must be imaginary 
until x becomes greater than h, after which there will be two 
branches of the curve, as shown by the double sign, alike situ- 
ated above and below the axis of X. Hence figure (a) will 
represent this curve, and the origin A will be an insidated point 
of the curve, because it is comprised in the equation as well as 
the various points in the two extended branches. 

The equation a^y^—x^-\-{h--c)x^-\'hcx=0, (3) 

is the same as (1) when we make c=0. From (3) we obtain 

V =-i-/ ^ (:g~ ^)(a^+c)\^ dy^^x''--2 x(h'—c)—hc 

\ ~a ) ' dx ^Jax(x—b)(xJ^) 

Now if we make x=0, or x=h, or a;= — c, either supposition 
will make ?/=0. 





c«) 


■ 


I^B 


H 


^H 



THEORY OF CURVES. 229 

Hence we have three points 
in which the ordinate is zero, 
in this curve. At A, fig. (b), 
when ar=0, at U when x=b, 
and at F when a;= — c. 

Every negative value of x 
less than c will give two equal 
values of y. Every such value 
of X greater than c will make 
y imaginary, and every positive value of x less than h will also 
make y imaginary, — hence figure (b) represents this curve. 

When c=0, vli^ becomes a point, and (c) 

the equation is represented by figure (a). 

When ^=0, and c retains its value, 
figure (c) represents the curve. 

When b=0, and c=0 at the same time, 
the loop AF must be taken off. 

Each of the values x=0, x=b, x-=. — c, 

reduces — to infinity, hence at the three 

corresponding points, figure (6), at A, at E, and at JP, the tan- 
gent is perpendicular to the axis of X. 

Solving the equation 2>x^ — 2a;(6— c)— 5c=0, will determine 
two real values for ar, and thus define the points at which the 
tangent will be parallel to the axis of X. 

We close this chapter with the following practical questions. 

1. Let the equaticm xy^4~^^ — ^ represent a curve \ has it any 
points of inflection? 

Ans. The points corresponding to ar= — , and 




y=zhj — ^a are points of inflection. 

2. The equation x'* — 3>^x^-\'B:^j=0, represents a curve ; has 
that curve any points of inflection? If so, designate them. 

Ans. It has points of inflection corresponding to each of the 

a __j ^T.„_„i. . 5a 

76 



points determined by making x=d= — -y and therefore l/=~ 



230 



DIFFERENTIAL CALCULUS. 



3. Has the curve represented by the equation a3y=x'* any points 
of inflection? 

\Ans. It has a double point of inflection at the origin 
( of the co-ordinates* 

4. Determine whether the curve whose equation is 

y=3x-|-18x2— 2x3 
has a point of inflection? 

{Ans. At a point corresponding to x=3, and conse- 
quently 2/= 11 7, is a point of inflection. 

5. Determine the point of inflection in the curve whose equation is 

'^ ax^ 




Ans. The point corresponds to y=|c. 



CHAPTER III. 

Osculating Curves. — Radius of Curvature. — !EvoIutes 
of Curves. 

CArt. 36.) The curvature of a curve is its deviation from a 
tangent ; and of two curves, that which departs most rapidly 
from its tangent, is said to have the greatest curvature. 

From this definition it is obvious that the greater the radius, 
the less the curvature, and our object is now 
to find the relation existing between the radius 
and the curvature. 

Let two circles touch each other internally 
at A. Conceive them to have a common 
tangent passing through A. Take any very 
small indefinite arc, as AB, and draw the 
chord AB, and the equal chord AB' to the 
other circle. The curvature of the inner 




OSCULATING CURVES. 231 

circle is measured by Am, and of the outer circle by Am', be- 
cause these are the relative deviations of these two curves from a 
tangent. 

Let r be the radius of the inner circle, and B that of the exte- 
rior and larger circle. Also, let c represent the chord AB, it 
will therefore represent the equal chord AB'. 

But the chord of a circle is a mean proportional between the 
diameter and the versed sine, therefore 

c'^=2r{Am), and c'^—2R{Am'), 

Whence r{Am)=R{Am'), 

which may be changed to the following form : 

Am : Am' : : _ : - 
r R 

That is, The curvature of two different circles varies inversely as 
their radii. 

(Art. 37.) A circle has the same degree or amount of cur- 
vature in every part ; but other curves, the ellipse for example, 
has different degrees of curvature corresponding to different 
portions of its circumference, and each small portion of any 
ellipse or any other curve may be conceived to coincide with a 
small portion of some circle. 

If a circle and a curve coincide at any particular point, it is 
an axiomatic truth that both the circle and the curve must have 
the same abscissa and ordinate corresponding with that point, 
and if the two curves coincide to any extent whatever, the first 
and second differential coeff dents of the circle will be equal to the 
first and second differential coefficients of the curve. 

The circle which thus changes its center and its radius to 
keep in coincidence with another curve, is called an osculatory 
circle. 

The equation of a circle is of the second degree, therefore it 
can have but two differential coefficients, and if we are able to 
express the radius of a circle in terms of the first or second dif- 
ferential coefficients of the co-ordinates, or by any combination 
of them, that radius will correspond to the circle which will 
coincide with the curve having the same variable co-ordinates. 



232 DIFFERENTIAL CALCULUS. 

(Art. 38.) The object of this article is to express the radius 
of an osculatory circle in terms of the differentials of the co- 
ordinates. 

In analytical geometry we found the general equation of the 
circle to be 

a and h being the co-ordinates of the center of the circle, and R 
the radius. 

Differentiating and dividing by 2 produces 

{x—a)dx+{y—b)dy=0. (1) 

Differentiating again, regarding dx as constant, we obtain 
dx--\-dy^+{y—h)d''y=Q, 

Whence (y-5)=_l^!±^). (2) 

This value of (y — b) put in (1) transposed, &c. and 

^ ^ dx\ d'y J ^ ^ 

Substituting the values of {x — a) and {y — h) as found in (2) 
and (3), in the equation of the circle, we shall have 

j.:,_dyW dx'+dy^ \,/ dx^-\-dy^ Y 
dx\ d'y J ~^\ d^^y J 



Or E.-.{dx-+dy') 

(dxd^yy 



3 
2 



Whence j;=_('^^-WF, 

dxd^y 

which is the general expression for the value of the radius of the 
osculatory circle. 

(Art. 39.) To show the practical utility of the preceding 
formula, we will apply it to the general equation of the conic 
sections, which is 

y^ z=2px-\-qx- . (Last eq. conic sections.) 

This equation, as we have before seen, will correspond to, or 
represent a circle, an ellipse, a parabola, or an hyperbola, ac- 
cording to the values assigned to 2p and q. 



OSCULATING CURVES. 233 

We can new obtain a general value for an osculating radius, 
which will apply to any of the conic curves whatever. 
By differentiating the last equation, we have 

dy^iP+^^^, (1) 

Taking the differential again, regarding dx as constant, and 
we have 

^g^_ qydx^—{p-\-qx)dxdy __ \qy^—{ p-\-q:r.Y \ dx'' .^. 

y1 y^ ' \ 

Whence R^ {dx^-^dy^Y _ ^\{v-\-<l^Y^tV\ 
dxd~y gy2_^p_^q^y 

By substituting the value of y^ in this last equation, we have 

j^_ ±z\(p+qxy+2px+qx'\' 



Or ;?-± \ {p+qxy+%px+qx'\ '^ ^ 



-p^ 



(3) 



The signs should be so taken as to render JR positive. 

This last equation expresses the radius of curvature for each 
and all of the conic sections, the origin being at the vertex of the 
major axis. At that point we have x=0. 

Whence R=Py for the radius of curvature at the vertex of the 
conic sections. For the parabola it is half the parameter. For 
the vertex of the origin of the ellipse, it is 

which is half the parameter of the major axis. 

The same value is found corresponding to the vertex of the 
hyperbola. 

For the parabola q=0, and if we assume p=l, what is the 
radius of curvature at the point corresponding to x=^l(i? 

Ans. QO.yVo- 

* If ^e find the expression for the normal of the curve whose equation is 
y^'—'ilpx-\-qx'^, and compare it with this equation, we shall perceive that 
the radius of curvature is equal to the cube of the normal divided by the 
square of half the parameter. 



234 DIFFERENTIAL CALCULUS. 

For another application, we require the radius of curvature 
for the ellipse at the vertex of the minor axis. 

For this point p= , g'= — , and x=A, 

Jx Ji. 

These values of ^, q, and Xy substituted in the equation, give 

A^ 

R = , wJiicJi is half the parameter of the minor axis. 

B 

To come more directly to the utility of the theory, we now 
require the radius of curvature of the meridians on the equator 
at the poles and at the latitude of 42°, taking the diameter of 
the earth as given by John F. W. Herschel, and the length of a 
degree at each of these latitudes. 

The equatorial radius is 3962.82 miles =A. 

The polar radius is 3949.68 miles =B. 

To find X corresponding to latitude 42°, we will make use of 
the mean radius 3966 miles, and subtract the cosine from the 
radius, and we obtain ir=1023 miles nearly. 

The radius of curvature at the equator is — =3936.26miles. 

A^ 
The radius of curvature at the poles is — =3976.2 miles. 

The radius of curvature in lat. 42° is found by the formula to 
be 3966.4 miles.* 

Hence, the length of a degree on the meridian at the equator 

is (3936.26)(3.1415962) _^„,^,^;„„ 

180 

la lat. 42° it is (3955.4)(3.1415962) ^gg ^3, ^^^^ 

180 

And in lat. 90° it is (3976.2)(3.1416962)^3, 3,, ^^^^^ 

180 

* To substitute for particular latitudes requires some care, as 9 is a frac- 
tion and negative. In this ellipse q= — 0.9933 nearly, J3=3936.26, 

(^4-5^:) 2 =16579991 .67, 2;5x=r.8052967.5, 9x2=— 1039507.13. 

Whence ^^(15540484.55)1^33^^^ 

(3236.26 2 



THE EVOLUTE CURVE. 



235 



(Art. 40.) An osculatory circle is one whose radius and po- 
sition of the center are in a continual state of change. 

Let M, M\ M'\ M'", &g. be points of a polygram inscribed in 
a curve. The perpendicular from to MM' is the first radius of 
the osculatory circle, and a perpendicular from the point 0' on 
to M'M" is the second radius of the osculatory circle, and so on. 
The points 0, 0\ 0", &c. if sufficiently near each other and 
properly connected, will form a second curve, which is called 

THE EVOLUTE CURVE. 



It is obvious from the figure, that the 
differences between two consecutive radii is 
0', 0' 0", &c. that is, the difference between 
the radii of curvature at any two points of the 
involute is equal to the space between the points 
of the evolute intercepted between them. 

In the last article, we have seen that the 
osculatory circle must correspond to the 
following equations : 

(a? — a)dx-\-{y — b)dy=0. 



Whence 



d'y 
dx\ d^-y J 




(3) 
(4) 

(S) 



And a=x-^(^^l+^ 

dx\ d^y 

The values of a and b correspond to the points 0, 0', 0\ and 
thus equations (4) and (5) will determine the evolute in any 
particular case. 

It is obvious in the last figure that the radius of curvature is 
normal to the involute and tangent to the evolute. 

As an example, let it be required to find the equation of the 
evolute of the common paraboloid; the equation of the involute 
is . y2=2par. 



236 



DIFFERENTIAL CALCULUS. 



This example requires the values of a and h deduced from 
equations (4) and (5), having 

dx 



nr)2 (7^2 

dy^z=z±. — : — , and d^y- 

y2 



(2) 



y y r 

Whence ^^l±^^=yJ^, and 6^=^, (1) 

And _^(^^-h^V+i^+^^ a=x+P+y- 
dx\ d^y J ^^^ p ~^~ p 

From the equation of the curve we have 

^= , and ^==2ar. 

P^ P P 

These values substituted in (1) and (2), will give 
8a;3 
~P' 



h^: 



(3) 



a=Bx+p. (4) 



From (4) we obtain x= — ^, and x^=S ±J-, which 

value put in (3), gives 

21p 
showing the law of connection between (a) and (b), or it is the 
equation of the evolute curve of the common parabola. 

Thus we might find the equation of the evolute of any other 
curve. 

Corollary 1. If we make 5=0, we shall have a=p, showing 
that the evolute of the parabola meets the focus. 

Corollary 2. If we make a less 
thanj9, 5will hQ imaginary, showing 
that the focus would then be a point 
of infection. 

Corollary 3. If we transfer the 
origin from A to 1), we shall have 
a'=ia — Pi b'=b. 

8a»3 




Then 



d'2=: 



27p 



POLAR CURVES. 



237 



Since every value of a' gives two equal values of h with con- 
trary signs, the evolute is symmetrical in respect to the axis of 
X. The evolute BF corresponds to the involute AMy and the 
evolute Df to the involute Am. 

The evolute of one-fourth of the ellipse is the diflference be- 
tween Jfi^and ADy that is, — - — — = the curve DF, along 
which the osculatory center moves. 



CHAPTER IV. 
On tbe differential expressions of Polar Curves. 

(Art. 41. J Before discussing spirals, it is necessary to de- 
termine general expressions for the arc, the secant, the tangent, 
the sub -tangent, &c. of a polar curve. 

The suh-tangent in polar curves is the part of th.Q perpendicular to 
the radius vector of the point of contact intercepted between the 
pole and the point where the tangent meets this perpendicular. 

Thus, Let A be the pole. Draw 

AMy Am, two consecutive radii, so 
near each other that Mm may be 
taken for the differential of the arc. 

Let MQ be perpendicular to Am^ 
then Qm is the differential of the 
radius. 

Draw AT perpendicular to Amy 
or parallel to Qm^ then, according 
to our definition, AT is the sub-tangent 
and MT the tangent to no particular 
arc, but corresponding to the polar 
radius AM, and the curve, whatever 
curve it may be. 

In respect to the differential, it is immaterial whether we con- 
16 




238 DIFFERENTIAL CALCULUS. 

sider the angle MAB, or the angle MA G, as the integral angle, 
the differential hh is the same for either. 

Let AM=r, and the angle MAX^=t. Take Ah—\y and hh=dL 
Then QM^=rdty Qm=dr, and if we put ds to represent Mm, 
the differential of the arc, the right angled triangle mQM will 



ds=Jr''dt''+dr'', 
for the differential of an arc in respect to polar co-ordinates. 

The differential sector AMm is measured by l(Am,QM). 
That is, ^(r-^-drydt. But as dr is comparatively nothing in 
respect to r, the limit of this product is 

r'^dt 

which is the area of an elementary/ sector. 

The similar triangles m^if and m^^give the proportion 

mQ : QM : : mA : AT. 
That is, dr : rdt : : r-^dr : AT. 

Passing to the limit, that is, taking r in the place of r-^r, 
we have 

2 a// 

AT—— — , the sub-tangent, 
dr 

The angle MAT being indefinitely less than mAT, we may 
regard MATaa a right angle, hence 



MT=^-AM'+Af' 



Or MT=r I\aJ2^> the tangent. 

As MN'is normal to the curve, the angle NMT is a right 
angle, and AM^ or r, is a mean proportional between AT and 
ANy therefore 

: r \ : r ; AN. 

dr 

Or AN= — , the sub-normal, 

dr 



TRANSCENDENTAL CURVES. 239 

In the right angled triangle ANM, we have 

That is, j\fM= I ^^^-Lyg the normal. 



CHAPTER V. 
On Transcendental Curves. 

(Art. 42.) Curves are generally divided into two classes, 
algebraic and transcendental, according as their equations con- 
tain purely algebraic or transcendental quantities. 

The curves hitherto examined, are algebraic; we now propose 
to illustrate and show some of the properties of some of the 
transcendental curves, beginning with 

THE SPIRALS. 

A spiral is a curve described by a point which moves along a 
right line in accordance with some fixed law, the line at the same 
time revolving uniformly about one extremity, its pole. 

When the motion of the point commences at the pole and 
moves uniformly over the length of the line while the line 
makes one revolution, the spiral then described will be the spiral 
of Archimedes. 

Thus, let AB be the line. A the pole. 
Let the point jlf commence at A, and when 
AB revolves to the position of A^, the 
point will be found at M, it having de- 
scribed the spiral curve AM in the same 
time. 

When AB revolves to the position of 
AD, the point will be at F, it having described the curve MP 
while the line changed from AN" to AD. 

When the point arrives at B, the line is in the same position 
as at first, and the s^ivalAMPB has been described. 




240 DIFFERENTIAL CALCULUS. 

Now let the line be indefinitely increased and the motion con- 
tinued, and an infinite number of revolutions might be made. 

To find the equation for this curve, let AB=a, the arc BN^=^ty 
and AM^=^r, the radius vector of the spiral at any point, as M. 

Then by the definition 

AM : ^iV=arc JB^^ : arc £JVDB. 

That is, r : a=t : 2rta. 

Whence r== — , ike equation of the curve. 

The transcendental quantity in this equation is t, the arc of a 
circle; hence, this curve is a transcendental curve. 

When t includes the entire circumference, it is equal to 2rta, 
and then the equation becomes r=a. When ^=4rta then r=2a, 
and so on indefinitely. 

THE HYPERBOLIC SPIRAL. 

(Art. 43.) While the line AB revolves about the pole, let 
the generating point move along the line in such a manner, that 
the radius vectors shall be inversely proportional to the corres- 
ponding arcs, then the point will describe the kyperlolic spiral. 

l.QtAB=a, AM=:l, A]Sr=r. 

The arc BN=t. 

Now from the definition we have 

AN : AM : : circ. BNDQB : arc BN. 
That is, r : \ '. '. 2a7t : t. 

Whence r= , the equation of the curve. 

Let AF=r, then we must designate the arc BD by i, and the 
proportion will be the same as before, and so on for any point in 
the curve. 

When AB has made one revolution, then^=2a7t, and the equa- 
tion becomes r=l, corresponding with the construction. 

The equation shows thatr cannot become zero until t becomes 
infinitely great ; that is, the spiral will meet the pole after an 
infinite number of revolutions, and therefore the minor revolu- 
tions may be compared to the whirling of a top. 




TRANSCENDENTAL CURVES. 241 

On the other hand, when t is very small, r will be correspond- 
ingly great ; hence, the curve, after passing N, will run off and 
become nearly parallel to AB, and in that sense AB is an asymp- 
tote to the curve, and hence the name hyperbolic spiral. 

The two preceding spirals, and indeed, all spirals that can be 
constructed or conceived of, are included in the general equation 

r=af, 

a representing a constant quantity, and n may be either positive 
or negative. 

When n is positive, the spirals will pass through the pole, for 
if then we make ^=0, we shall have r=0. 

In the spiral of Archimedes ?i=], and in the hyperbolic spiral 
>?= — 1, as we have just seen. 

LOGARITHMIC SPIRAL. 

While AB revolves uniformly about the pole, let the gene- 
rating point move along the line AB in such a manner that the 
logarithms of the radius vectors may be proportional to the 
measuring arcs, it will describe the logarithmic spiral. 

From this definition we have at once 

t=\og.r, 

f<yr the equation of the logarithmic spiral, in which r represents 
the radius vector, and i the measuring arc. 

(Art. 44.) We can now deduce some of the properties of the 
spirals by the application of the differential 
expressions for the polar curves, as deter- 
mined ih the preceding chapter. 

Let mMh be a portion of a spiral curve, 
the radius vector AM=^t, and the equation 
of the curve 

rz=af, 

it is required to determine AT, MT, MN^ 
and AN. 




242 DIFFERENTIAL CALCULUS. 

In (Chap. V,) we found 

Sub-tangent AT= , and because r=afy we have 

dr 



Or 



dr=naf~*di. 
dt_ 1 
'dT'naf-^' 



r^dt CL^t^^ dt^^ 

Whence = .= , the sub-tangent, and when 

dr nat"^^ n 

»=1, as it is in Archimedes* spiral, the value of the sub-tangent 
is ai^ . But in that spiral T=^at, and a= — 

Whence at^ z=.—='i7^r^ , 

a 

If r=l, the sub -tangent will be 2rt, the circumference of the 
measuring circle, and after two revolutions, it will he four times thai 
length, and so on, as the squares of the number of revolutions. 

This property was discovered by Archimedes. 

In the hyperbolic spiral n=^ — 1, the corresponding sub-tangent 
is then — a, a constant quantity. 

The tangent MT=^r li\^^_^=r jT\4^=afJ\J^^. 
^ ~ dr^ 



The normal MjV^= M^"" JUr^== l—Jl-\-n^aU'^'^''. 

The sub-normal AIf= — = 

dr naf~^ 

In the spiral of Archimedes the sides of the triangle MA, 
AT, and TM, are in the proportion of 1, ^, and tJl-^-t^ . 

(Art. 45.) As the angle MAT is a right angle, we have 
MA : AT : : radius : tsm.AMT. 

That is, r :^yj^ : : 1 : tan. AMT. 



Whence tan.^ifr=^^ 



dr 

¥T=: 

dr 



TRANSCENDENTAL CURVES. 243 

Let us apply this to the logarithmic spiral, the equation of 
which is 

t=\og.r. 

Whence dt=m — , 

r 

m being the modulus of the system. 

Therefore tsiii.AMT=^—=:m. 

dr 

That is, the angle between the radius vector and the tangent 
to the spiral at the point of contact is constant , and its trigonome- 
trical tangent is equal to the modulus of the system. If i is the 
Naperian log. of r, the angle will be 45*^. 

(Art. 46.) A logarithmic curve is not necessarily a spiral, for 
it is obvious that if we take rectangular co-ordinates and assume 
one ordinate to be a number, and the other a logarithm of that 
number, we shall thus have an equation which will produce a 
logarithmic curve. 

The logarithmic equation is y=^a^, and taking x for the ab- 
scissa, and y the corresponding ordinate, the equation will mark 
out a curve, and a particular curve when a is given. 

As is well known a is the base of the system, and x is the 
logarithm of the number y in that system. 

We must also recollect that a cannot be 1, for every power of 
1 is 1, and in that case the variations of x would produce no 
variations in y. 

Let M'BMhe the logarithmic 
curve whose equation is y=a', 
and make x=0, then we shall 
have 

y:=a°=l=AB, 
and this will be the value of AB, 
whatever be the value of a, show- 
ing that all logarithmic curves will cut the axis of Fat the dis- 
tance of unity, whatever be the system. 

Let a be greater than 1, and x positive, then y will be positive 
and greater than 1, corresponding with the figure AF and FM. 




244 DIFFERENTIAL CALCULUS. 

When X is large, a small variation in x produces a much greater 
variation to y. 

When X is negative, the equation becomes 

showing 2/ to be a fraction, or less than unity, but y cannot be- 
come zero until x becomes infinite and negative, showing that 
the curve will meet the axis of T^at an infinite distance to the 
left of the origin. Hence AP' is an asymptote to the curve, and 
the curve itself can therefore be classed with the hyperlolas from 
whence comes the term hyperbolic logarithms. 

The equation of the curve is y=a^ . 

Whence \og.y=zx\og.a. 

^=y\og.a. (1) 

But — represents the tangent of the angle which the tangent 
dx 

line forms with the axis of JT, hence that tangent will be parallel 
to the axis of X when 2/=0, and perpendicular to it when y is 
infinite. 

But the most remarkable property of this curve is its sub- 
tangent, represented by the symbols fy — V (Art. 24). 

dx 1 

y — = 

dy log. a 
That is, the sub-tangent is a constant quantity y and equal to the 
modtdus of the system, whichever system that may be. 

(Art. 47.) Another important transcendental curve is 

THE CYCLOID. 

The cycloid is a curve described by the motion of a point in 
the circumference of a circle while the circle rolls along a right 
line, the point commencing to move from the line, and to make 
the curve complete, it must meet the line again, during which 
time the circle will make one revolution along the line. 

Another revolution, and the point will describe another cy- 
cloid, and so on indefinitely. 




TRANSCENDENTAL CURVES. 246 

Let M be a point in the circle 
BMR, and conceive it to roll along 
the line RO from R to G. The 
circular arc RM falls down upon 
and measures RA, and the point 
M moves over and describes the 
curve MA in the same time, and 
this curve MA is a portion of a cycloid. 

To find the equation of this curve we must determine the re- 
lation between AP and PM. 

Conceive A to be the origin of the co-ordinates, and put 
AP^x, PM=y=RE. 

Let the radius of the generating circle be r, and the arc MR, 
the radius unity, be z, then the value of the arc MR will be rz, 
which is equal to AR. 

Now AP=AR—ME, 

That is, x—rz~ME. (1) 



But ME=:^ J BE, ER= J{2r—y)y. 



Whence a;=arc(sin.=^2ry — y^) — ^2ry — y^. (2) 
If in this equation y be taken negative, the value of x will 
become imaginary, showing that if can never pass below the 
line AR. When y=0, x will equal an arc whose sine is 0, 
hence x will equal also. When y=z<2,r, x will equal the arc of 
1 80° to the radius of r; y cannot be greater than 2r, for then x 
would become imaginary, showing the absurdity of any such 
hopothesis. 

(Art. 48.) But the properties of this curve are most easily 
deduced from its differential eqxmiion. 

To find the differential equation of this curve we will differ- 
entiate (1), which is 

x=^rz — sin. {rz). 
dx=^rdz — cos.{rz)dz. (3) 

But sm.(rz) = j2ry — y-. 

Whence eos,(rz)dz=Si:^^. (4) 

J2ry—y^ 



246 DIFFERENTIAL CALCULUS. 

But COS. (rz) is OH, which is equal to (r— y), therefore 

rdz=-jM=.. (5) 

Values taken from (4) and (5), and substituted in (3), will 



dx 



_ rdy 



Or 



_^ , {r—y)dy 

*J^ry—y^ J^ry—y^ 



J%ry—y^ 
which is the differential equation of the cycloid. 




(Art. 49.) Now by 

the application of (Art. 
24), we can readily 
find expressions for the 
tangent, sub -tangent, 
normal, and sub -nor- 
mal of this curve 



The tan.^7'=y^|!+l=y^^. 



2r — y 



r 



Sub-tan. TF=l^-^=-. — . 

^y J2ry—y^ 

Normal MN=^l- Jdx'^+dy^ = J~^- 
dx 



ydy. 



Sub-normal PN=il'"A=: J2ry—y^ , 
dx 

These values being determined,on the greatest ordinate BD, 
describe the generating circle. Take any point in the curve, as 
My and draw ME parallel to AP. Join BE and ED, PM is 
parallel and equal to BE, each equal to y. 

Now BD=^^r, and by the property of the circle EE^=^ 

J^ry—y^. 



Now PiVand EE are equal, since each is equal to J2ry — y^, 
and the two triangles i/PiY and EEB are equal, whence MN= 
HB and MNBE is a parallelogram. 



TRANSCENDENTAL CURVES. 247 

Because MN is normal to the curve, and TMm a tangent at 
the point M, the angle NMrn is a right angle equal to the angle 
BHD in the semi-circle, and as A'JUf and BH are parallel, it fol- 
lows that Mm, the tangent to the curve at M, is parallel to the 
corresponding chord of the generating circle described on the 
greatest ordinate. 

(Art. 50.) Resuming the differential equation of the curve, 

J^ry — y^ 
Placing it in the form 

^_ J9.ry—y'^^ y2r__ 
dx y V^ 

Making y=0, we have — =infinity, and making y=2r, we 

have -^=0, showing that at the point A, a tangent to the curve 
dx 

is perpendicular to the axis of X, and at the point D a tangent 
is parallel to the same axis, 

(Art. 51.) To find the radius of curvature at any point, as 
My we must apply the general equation, (Art. 38), 

dxdy^ ^ ^ 

As the second differential of dx is not required by the formula, 
we may regard it as constant, therefore the differential of 

dx^-^^I— 



J^ry—y^ 

is 0={yd-y+dy-)JW^y^---^^^k^^^* 

J<iry—x/ 

Reducing, and 

0=(2ry— 2/2)c?2y-f.r^y2. 

rdy^ 



Whence d'^y-. 



i--y^ 



* The denominator of this differential is omitted for obvions reasons. 



248 DIFFERENTIAL CALCULUS. 

Substituting the values of dx, dy, and d^y, in (1), we have 

R-( y'^y' ^^y^y y C^ry-y -)- ^{9.ry-y-) 
\%ry—y^ J ydy rdy^ 

Or i2=_(?!Z.fe 

{2ry — 2/2 y 

3 

Or Ii=i?Il}l=2j2^i. 

ry 

That is, the radius of curvature at any point, as M, is double 
of the corresponding normal MJV, The radius of the curvature 
at A is therefore zero, and at D it is twice DB, or 4r. 



vi^ry-y^Y 
,2 rydy^ 



THE EVOLUTE. 

(Art. 51.) In (Art. 40), we find the two following equations 
in which the quantities a and b represent the co-ordinates of the 
center of the osculatory circle; their relation, or one in terms of 
the other, will give the equation of the e volute. 

(dx^-\-dy') 



f—h-. 



d'y 



dx\ d^y / 



The values of dx, dy, and d^y, substituted in these equations 
and reduced, the results will be 

Whence 



and X — a= — 2j2ry 
y= — hf and ;r=a 



-y^- 



2j2ry — y^. 

The first of these equations 
show that QM'z=PM. The 
last article demonstrated that 
MN=NM', therefore the two 
triangles PMN and NQ,M' are 
equal. 

PN, the sub-normal, is equal 
to NQ. But this sub -normal 
is J'iry — y^ . Hence, PQ equals 2^2ry — y"^ , which subtracted 
from A Q, (a), gives x corresponding with the second equation. 




TRANSCENDENTAL CURVES. 249 

If the value of x and y be taken from the last equation and 
substituted in the geometrical equation of the cycloid, we shall 
have, after a little reduction, 



a=arc(sin.=(7— 2r6— ^2)— ^— 2r6— 62, (1) 

the equation of the evolute AM'A'. 

To make this equation more clear, we will transpose the origin 
from A to A\ BA' being equal to 2r. 

Take A'P'=a\ and P'M'=h', then it is obvious that 
a=AB-^A'P'=:7ir—a'. And as QM'=^J), andP'§=2r, 
we shall have — &=2r — h'. 

Substituting these values in (1), we have 



rtr — a'=arc ( sin . = J2rb'—b' ^ ) + J2rb' — b' 



Whence a'=z7tr — arc(sin.=^2r5' — b^ ) — J^rb' — b' 



But 7tr — arc(sin.=^2r6' — 6'2)i=arc(sin.=^2rJ' — b'^ ). 



Hence a'—arc(sm,=j2rb'—b'^)—'j2rb'—b'^. 

This equation has the same form and contains the same con- 
stant (2r) as the equation of the cycloid, hence the curve A'M'A 
is also a cycloid equal to the primitive one, — or the involute and 
evolute are equal. 

OTHER GEOMETRICAL DIFFERENTIALS. 

(Art. 53.) The dififerential of a plane geometrical surface is 
obvious, as the adjoining figure will illustrate. 

Let A be the origin of co-ordinates, 
AF=x, and PM=^y, and they may be 
regarded in reference to the rectangle 
PBy or to the triangle AMP, or to 
the curve NMP. 

Let AP or x receive a small incre- 
ment dx, then {ydx) is the differential 
•parallelogram, which is the differential 
of the parallelogram BP^ or of the triangle AMPy or of the 
curve surface MNP, according to the given relations between x 
and y. 




tm DIFFERENTIAL CALCULUS. 

(Art. 54.) If we conceive the surface ABMP to revolve on 
tIP as an axis, it will then describe a cylinder, and the differential 
parallelogram (ydx) will describe the differential of this cylinder, 
which will be measured by {rty^dx). 

The revolution of the triangle AMP will describe a cone, the 
differential of which will also be represented or measured by 
[jiy^dx). Also, the revolution of the curve surface NMP^ on 
the same axis, will describe a segment of a circular solid, the 
differential of which is measured by the same expression 
{Tty^dx). 

Let the reader bear in mind that different, integrals may have the 
same differential , as articles 63 and 54 illustrate. 



(Art. 55.) Observe that Jdx^-\-dy^ is the differential of a 
line which is either a straight line or a curve, according to the 
relative values of x and y, and the revolution of this line on the 
axis of Xwill describe the differential of a surface, the surface 
of a cylinder, or of a cone, or of a curved surface, as the case 
may be. 




THE INTEGRAL CALCULUS 



CHAPTER I 



(Art. 56.) The integral calculus is the converse of the dif- 
ferential, and all our rules of operation refer back to the same 
general principles. 

Although the operations are the converse of those in the dif- 
ferential calculus, we must not infer that they are equally 
obvious, and one as easy as the other. 

To cube a number, and to extract the cube root of a number, 
are converse operations, and the last can be deduced from the 
first, but it requires far more care and attention. 

Without farther remark we will proceed with the subject, 
commencing with the most simple case. 

The differential of a simple quantity, as x, y, z, or any other 
single symbol, is expressed by writing the sign d before it, as dz, 
dy, &c. Hence, to pass from the differential quantity to its 
integral, we simply remove that sign, and for dx write x. To in- 
tegrate dx-\-dy — dz, we simply write x-\-y — z, &c. &c. 

The differential of x^ is ^x^dx. 
The differential of x^ is 5x*dx. 
The differential of 2/" is my'^-'^dy. 
Now therefore, to integrate an exponential quantity consisting 
of a single term, we must frame a rule that will change ^x^dx 
torc^, Sx'^dxiox^, and my^~'^dy to ?/"*. 

These operations can obviously be performed by the following 

Rule {A). Add one to the exponent, divide hy the exponent, so 
increased, and take away the differential factor, 

251 



252 INTEGRAL CALCULUS. 

EXAMPLES. 

1. What is the integral of Ax^dx? Ans. ^x^ . 

2. What is the integral of ly^dy? Ans. ^^y^, 

3. What is the integral of SOP'^dP? Ans.eOjPl 

4. What is the integral of ^x^^dx? Ans. -y-^ x'^^K 

3(m-|-l) 

5. Integrate the differential _^_f. Ans. — — . 

x^ x^ 

adx „ 

6. Integrate the differential — — j. Ans. -___ . 

3a;3 V^ 

(Art. 57.) The diflferential of an equation like 

y—ax'\-h, (1; 

is dy=adx, and the attached constant h disappears. 

Now take the reverse operation, and pass from the differential 
to the integral, and we have 

y=ax, (2) 

and the constant h is lost, and thus it might be in any other 
ease — hence an integral obtained from a given differential may 
require a correction, which it is customary to denote by the sym- 
bol C. 

This being the case, the integral completey or equation (2), is 

y—ax-^-C. (3) 

To determine the value of C we must know the import of 
equation (1), or as most writers express it, we must Tcnow the 
nature of the problem, or the relation between the variables at 
some particular point. 

Now from analytical geometry we know that equation (1) is 
the general equation of a straight line, and when ar=0, y must 
equal b. Making this supposition in (3) we have 

This value of C put in (2), and we have (2/=aa;-f-^), equation 
( 1 ) completely restored. 



INVESTIGATION OF RULES. 263 

Again, let us examine the first example under rule (A), 4x^da 
might have been the differential of (|>'c«+a), or of (^^x^-}-m), or 
of any other constant attached to the variable part. Hence, it 
is very proper to designate fa;« as the partial integral, and 
(1^^ + ^) ^^ the complete or the corrected integral. 

After we determine the value of C, the result is called a. par- 
ticular integral. 

We cannot determine the value of C without some given or 
known conditipn^ but with a condition it is very easy. 

Thus, take the last integral (|a;^-j-(7), and suppose the value 
of the whole must be 4, when a?=l, then |-J-(7=4, and (7=3^, 
and the particular integral is - -^' 

(Art. 58.) Rule (^4) fails in one particular case, as in the 
following example : 

Required the integral of a^'^dx. 

^^-1 + 1 I 

By the rule -'- =_= infinity, 

^ —1+1 ^ ^ 

But this is incorrect, for x'^dx is — , which is the differential 

X 

of the logarithm of x, (Art, 10), and therefore the integral is 
(log.a;-|-(7), and as the logarithm of a; is not an algebraic quantity, 
the rule failed. 

(Art. 59.) When we wish to indicate an integral we use the 
symbol T, which is a prolongation of the letter S, the initial of 
the word sum, as the integral was conceived to be the sum o/ , 
a great multitude of minute differentials. ; 

Thus, if we wished to convey to the mind the integral of 
mx^dx, we simply write fmx'^dx, and so on for any other 
quantity. 

Constant factors may be written without the sign. Thus, 
fa.x'^dx is the same as afx'^dx. 

The more general index for this is afXdx, X being a symbol 
indicating any algebraic function of x. 

(Art. 60.) In general, the constant is arbitrary, since what- 
17 



264 INTEGRAL CALCULUS. 

ever value be assigned to it, it will disappear in taking the dif- 
ferential. This arbitrary nature of the constant fortunately 
enables us to cause the integral to fulfil any reasonable condition. 

For example, we require that the integral of bx'^dx shall be 100, 
when x=a. 

The integral complete is 

/5^'<'^=^-+(?- (1) 

Now by the condition, when we write o in the place of x, the 
second member of (1) is 100. 

That is, ^1+C=l00. 

Or C=100— --. 

4 

^-|-100~r^ Y the 

integral required. 

If in (1) we make x=a, and then x=by the expressions may 
be generally expressed thus : 

JXdx=A+C. JXdx=B+a 

x=a ar=s6 

Whence by subtraction, 

JXdx—JXdx=B—A. 
a:=6 x=a 

This indicates that the integral has been taken between the 
limits a and b, and it is usually written 

jlXdx=B^A. 

the subtractive integral being placed below. - 

EXAMPLES. 

1. Find the integral of Sx^dx between the limits of x=l and 
x=3. Am. 120. 

Jl Gx^ dx=^3.^^= 1 20. 

2. Find the integral of 7x*dx, between the limits of x— — 1 and 
x=2, Ans. 21. 

J^lx^dx^lx'^^C, 



PRACTICAL EXAMPLES 266 

(Art. 61.) Many binomial differential expressions may be 
reduced to monomials by algebraical artifices, and then inte- 
grated by Rule (A) as illustrated by the following 

EXAMPLES. 

1. Let du=(a-\-bx')'^cx''-^dx 

be a differential equation, the integral of which is required. 

Let the learner strictly observe its form. The exponent of the 
variable x within the parenthesis is w, without the parenthesis it 
is n — 1, one less. 

When this is the case, expressions in the above form are alioays 
integrable by the following process : 

Place 65+&k"=2. 

Then rib^-^dx^dz, x'^'^dx^—, 

nb 

Substituting these expressions in the equation, and we have 

du^=^—-z^dz. 
no 

Integrating by the rule, 

nb{m-[-\y 
And replacing the value of z, we finally have 

nb[m-\-\) 



2. If du^il+Zx^YQxdx, «=il+??!il+(7. 

3. If du=:^—{^—^x^)^x^dx, u=i^^^^-l~+C, 



8 



4. If du={a-\^x^Yxdx, w==iHh^?!ll+a 



266 INTEGRAL CALCULUS. 

xdx _^ 1 



7. If t/w=_^'^, w=a- ^ 



(Art. 62.) Differential equations of the form represented in 
the last article, are always integrable when mis a whole number, 
whatever may he the relation of the exponents within and without the 
parenthesis, or whatever may be the number of terms within the pa- 
renthesis. 

For example, if we require the integral of the differential 
equation 

du= (a-\-bx-{-cx^-\-&G) "^exdx, 

and if m is a whole number, we can expand the quantity in pa- 
renthesis, and then multiply each term by the part without the 
parenthesis, and we shall have a series of monomials, each one 
of which can be integrated by Rule (A). 

This being understood, the integration of the following differ- 
entials can readily be obtained. 

1. Given du^=(a-\'bx)^xdx tofndvL. 

By expanding du=(a^-]-3a^bx-\-Sab^x^-{-b^x^)xdx 

—a^xdx-\'3aHx^dx-\^3ab^x^dx-\-b^x*dx. 

^, a^x^ , Sa^bx^ , 3ab^x^ , b^x^ . n 
Whence «= +-^ 4- 4- +C^. 

N. B. We add but a single constant, for if we add to each 
term of the second member, the sum of them would be a con- 
stant quantity, which might be represented by C alone, hence, 
a single constant is all that is required. 

We may also integrate the last example, and others similar to 
it, as follows : 

Place a-^x=^z. Then dx^—, and a?=i:Z?. 

ft h 

Whence xix^'^'-'^l, and du=.^-^-"£^. 



PRACTICAL EXAMPLES. 



05 az* 



By investigation «= — -|-(7. 

Replacing the value of s, and 

2. Gfiven dzi={l — cuc'ybdx to find m. 

■ . ■ 2rt5a;"*» , a^bx^"^^ .rr 

Am. u=ox — + +0. 

nJ^\ ^ 2/1+1 

Given du=z( —-\-x\ x^dx to find u. 



<H' 



Ans, uz=\og.x+^+^+^+C. 

(Art. 63.) Every equation in the form 

c?2^=^ic'"(a+5a;)"c?a;, ( 1 ) 

can be integrated when either m or n is a whole positive number, 

Ist. Let m be a whole positive li.umber, and n fractional or 
negative. 

Place a+5a?=0, then t?«:=^, ^n.^(g— «)" 

b 5° 

Whence du=,^{z—a)'^z''dz. (2) 

Now as m is a whole number, this last equation can be inte- 
grated by (Art. 62.) 

2d. Let w be a whole positive number, and m fractional or 
negative, and (1) corresponds at once to (Art. 62.) 

EXAMPLES. 

1. Integrate the differential du=x^(a'\'bx^ydx. 

Place a+6a;2=2. Then xdx=^ — , a;^= 

^ 26 b 

2b^du=z^dz—az^dz. 



258 INTEGRAL CALCULUS. 

^63 16 ' 

Ans ,,^ (3g-5a) J_ (Shx--2a)(a+bx^)^ 
\5b^ 1662 

2. Integrate the differential du=2x(l — Sxy^dx. 

3. Integrate the differential du= 

X — 1 

Ans. w=^+a?— |-flog.(a;— l)+a 

4. Integrate the equation dy= ^ '^_ ^ — . ( 1 8th Ex. Art. 6. ) 

Ans. y=(a-{-Jxy-^C. 

5. Integrate the differential 

du^(?^l±^^^. (16th Ex. Art. 6.) 
Va2+aj2 

This example may be put in the following form : 

Ans. u=^2xJa^^\^+C. 

6. Integrate the differencial du—^^ ~^^^^ ^. (13th Ex. Art. 6.) 

2Vl—a; 

Ans. w=(l-|-a!)^l — x-\'C. 

(Art. 64.) We have seen in (Art. 10) of differential calculus 
that the differential of the logarithm of a number is the differen- 
tial of the number divided by the number, multiplied by the modulus 
of the system. When the modulus is one, the system is the 



PRACTICAL EXAMPLES. 269 

hyperbolic or Naperian, and the constant disappears, or it is not 
written. A unit factor is not visible. 

Hence, whenever we observe differentials in the form 

^, or -^, or l^.?^f, or (^+to).fa^ 
X a-\~x ' ax-\-x'^ a-\-bx-\-cx^ 

we know that the integral must be the log. of the denominator, 
plus a constant. 

Thus the integrals of the above expressions are 
wlog.rr+C, \og.(a-\'x)-{-C, \og.{a-{-bx^cx^ )-\-C, &c. 

The logarithmic form of equations, or of differential expres- 
sions, is not always apparent in consequence of constant factors, 
but the form can be made apparent by a little algebraic artifice, 
as the following examples will illustrate. 



1 . Integrate the differential 



EXAMPLES. 

5x^dx 



I5x'+2l 

Put I5x'+21=z. Then 60x^dx=dz, or 5x^dx:=~. 

12 

Whence f^^-= fl.^^l rl^=llog.( 15^^+21 )+C 
-^ 15a;^+21 *^ 12 ^ 12^ ;2 12 ^ ^ J^ 

2. Integrate the differential equation du=^ ^ ' ^i— . 

2y+2/2 

Ans. w=log.(2y4-y2)-)-C. 

3. Integrate du—^ — ^. Ans. u= — flog.(l — y^)-{-C. 

4. Integrate rfw=l_Z!L_i'z^lL. 

^ 3+2y2_3y3 

Ans. u=2\og.(3-\-2y^—3i/^)-\-C. 

In the application of this branch of the science a sufficient 
number of examples will occur to exercise the student in loga- 
rithmic functions, and therefore we give no more at present. 



260 INTEGRAL CALCULUS. 

(Art. 65.) We have seen that the differential of a product 
as xy is (ccdy-\'ydx) . 

Therefore, the integral of (xdi/-\-ydx) is xy, and this must 
aerve as a fundamental rule for integration, and we now propose 
to show that this harmonizes with rule (-4). 

The differential expression obviously contains two variables, 
because we have dx and dy ; hence, the integral will contain x 
and y, but how connected, or how related, we are not supposed to 
know, at the present moment. 

But X must be equal to, or greater, or less than y. Let a be the 
difference between them, and that difference is constant. 

That is, xz:^y±ia. (1) 

Whence ydx=ydy. 

But xdy={y-±ia)dy. 

du-=xdy-\-ydx=9,ydy±iady. 

Now by integration, 

u=J^(xdy-{-ydx)=y^ ±:ay= (y±:a)y=^xy, Ans. 

Again, we can assume x=ay, in which a is greater, equal to, 
or less than one, as the case may demand. 

dx=ady, ydx=aydy, xdy=aydy. 

rdu=r(xdy-\-ydx) z=r2aydy=ay^ =ay.y=xy, A?is. 

Thus we perceive that either operation corresponds to rule (A) 
after the transformation is effected. 

(Art. 66.) We may take another view of this case. When 
we differentiate a product like xy, we conceive one letter, as x, 
constant, and the other variable, and thus we obtain xdy, the 
partial differential. 

Then we conceive x to be variable and y constant, and under 
that supposition we obtain the other partial differential (ydx.) 

Now if we take either of these partial differentials and integrate 
ou the supposition that the letter having the sign d prefixed is 
the variable one, and the other constant, and we obtain xy for 
the integral, and if we take each partial integral and integrate, we 
get xy twice, or 2xy, hit we must take hut one of theinfor the in- 
tegral, for obvious reasons. 



PRACTICAL EXAMPLES. 261 

The same principle holds good in relation to the three or more 
letters. The differential of xyz is 

xydz-\-xzdy-\-yzdx. 

Now if we integrate each of these expressions on the suppo- 
sition xy is constant in the first term, xz constant in the second, 
and yz constant in the third, we shall have 
xyz-\-xyz-\-xyz. 

Here are three ^qual integrals, but we must take but one of 
these for the whole integral, because the diflferential was effected 
by three distinct suppositions. 

This principle liolds good in relation to a quotient, as -, the 

diflferential of which is 

ydx—xdy^ ^j^.^j^ ^^^ ^^ written ——xy-^dy, 

y"" y 

Integrating each of these expressions on the supposition that 
y is constant in the first, and x constant in the second, we have 

y y 

but we must only take one of these for the integral, for the same 
reason as before. 

We may also change the form of this differential by substitu- 
tion, so as to make rule (^A) applicable to it. 

Thus place f?w=:^^=^. (i) 

Now put y^=tx, t and x being variable, for if t were not varia- 

X 1 X \ 

ble, the fraction -, or its equal — =-, would represent only a 

constant, which could have no differential, and therefore / as 
well as X must be variable. 

If y=tx, ydx=txdx, and xdy=txdx-\-x'^df. 

Whence du=^^J^=-^'^=-t-^ dt. 

y^ . ^2^2 



Integrating by rule (A)^ and we have 

u=-, but _=- 

( t y 



262 INTEGRAL CALCULUS. 

EXAMPLES. 

1. Integrate (6xy — y^ )dx-\-(3x^ — ^xy)dy. 

Ans. Sx'y — y^x. 

We integrate the first part on the supposition that y is con- 
stant, and the second on the supposition that x is constant, and 
we obtain 

Sx^y—y ^x-^Sx^y—y^x, 
and because we make two distinct suppositions, we divide by 2. 
Then test the result by taking the differential. 

Again we may integrate the last example as follows : 

Place du=(6xy—y^ )dx^{Sx^ —2xy)dy, ( 1 ) 

and assume x=ay, then dx=ady, and (1) becomes 

du—(6a^y^—ay^)dy+(3a^y'''~2ay^)dy=9a'y^dy—3ay^dy. 

Whence u=Sa^y^ — ay^=Sa^y^.y — ay.y':=3x^y — xy', Ans. 

2. Integrate the differential equation 

du=(2y'' x+3y^ )dx-{-(2x^ y+9xy^ -\-8y^ )dy. 

Ans. u=x^y^-\-3xy^-\'2y^+C. 

3. Integrate du^ i^' ^f >dx+{-' +-' )ydy , 

J(b'^+yn(a-i-x-) 



Place ^62+y3=P, and Ja^-\-x^ = Q. 

The several members will then reduce into the diflferential of 
a product. (See the author's Sequel, page 342.) 



Ans. u=Jb'-+y\Ja^-{-x^ + G. 

4. Integrate du= ^-^ 

Place (a—y+z)^=P. Ans. u=P^+C. 

5. Integrate du=6xdy-\-6ydx-\'3bdy-\--2cdx. 

Ans. u—6xy-\-Sby-{-2cx-\'C. 
Or u=(2x+b){Sy+c). 



CIRCULAR DIFFERENTIALS. 263 

6. Integrate examples 1, 2, 3, (Art. 3). 

7. Integrate efw=?(2^^^^II^). (This is Ex. 15, Art. 6.) 

3 
Place x=a'i/. Ans. u= — -{-C. 



CHAPTER II. 

On the Integ^ration of Circular Differentials. 

(Art. 67.) In (Art. 9), we have seen that if u designates 
an arc of a circle, and x its sign, we shall have, (radius being 
unity), 

J dx 



Whence f- ^^ ^u+C, (1) 

J\—x^ 
We know however, that when the sine of an arc is 0, the arc 
itself is 0, or 180°. Regarding it as 0, equation (1) becomes 
0=0+ C, or (7=0. 
Hence the whole integral is the arc of a circle whose sine is x, 
which is sometimes written 

f — -=arc(sin.=a;). (2) 

Jl—x^ 
When we can integrate the first member of this equation in 
numerical or algebraic terms, we shall then have the numerical 
value of the arc of the circle, to compare with the numerical 
value of the sine. 

(Art. 68.) When u is an arc and y its cosine, (radius unity,) 
we have 

(^t^=-> "^^ - . (Art. 9.) 



264 INTEGRAL CALCULUS. 
Whence f— ^JL. =u+a (3) 

To determine the constant we must take a particular case. 
Estimating the arc from the commencement of the first quotient 
the cosine of zero arcis radius, and from thence the cosine dimin- 
ishes and becomes 0, when the arc becomes a quadrant. Hence 
when 7/ is 0, the first member of equation (3) is ^7t, and u must 
also equal ^h; therefore 0=0, and the entire integral is 

f — — --^— =!t<=arc(cos.=y). (4) 

Again, let u be the arc (always less than 90°, and estimated 
from the commencement of the first quadrant), and t the tangent 
of the same arc, then w and t will commence and vanish to- 
gether, and integrals connecting them will require no correction. 

Now from (Art. 9), we have at once 

/-^-=='M=arc(tan.=0- (5) 

When u is an arc estimated as above, and v its versed sine, we 
have 

r — ;__^__>=?^=arc(vers. sin. ==«;). (6) 

(Art. 69.) It frequently happens that we have expressions 
to integrate in the form 

dx 



All such expressions indicate a circle, whose radius is a in 
place of unity, and x represents the sine of an arc if the expres- 
sion is positive, and a cosine if it is negative. 

In the above expression, if we suppose x represents the sine 
of an arc, and a the radius of the circle, and if we take z to 
represent the sine of the same arc in the circle of radius unity, 
we shall have 

a \ x : \ \ : z. 

Whence ^=f, dz=—. 



Therefore 



CIRCULAR DIFFERENTIALS. 266 

dx adz dz 



And r = r— ^ — =arc(siD.=g)=aicAiD.=:^\ 

J Ja^—x^ ^ J\^z^ \ a J 

the arc. still taken in a circle whose radius is unity. 

From this we may summarily conclude that the integral of 

— — — ^ is an arc whose cosine is ^. 

(Art. 70.) We have just seen that the integral for a tangent 
to the circle of radius unity is ^asf? .si; 

df 

But suppose we have before us tlie expression , to be 

integrated, we would examine and see if it were not the differ- 
ential of a tangent to a circle whose^ radius is a, and if so, 
determine its integral. 

Let t be the tangent of an arc, and a the radius, and z be the 
tangent of the same arc to radius unity, then 
a \ t \ : \ '. z. 
i=az, and dt=ad8. 
dt adz _\/ dz \ 

=_arc('tan.=^. ) 



Whence 



Or f-J^-=- arcAan.=i\ 



the arc being estimated to the radius unity. 

In view of the foregoing, and on inspecting equation (6) of 
(Art. 68), we will venture to conclude that 

C — — = arc( versed sin. =!!.). 



266 INTEGRAL CALCULUS. 

CHAPTER III. 

INTEOR ATION BT SERIES. 

Integ^ration of Rational Fractions. 

(Art. 71.) Expressions in the form Xdx, in which X is any 
algebraic function of x, and which cannot be integrated by any 
of the preceding artifices, the quantity Xmay be expanded into 
a series of simple terms, and then we can multiply each term 
by dx, and integrate. The sum of the integrals so found will 
be the approximate integral of Xdx, provided the series is con- 
verging, and is carried to a sufficient number of terms. 

When the series is not converging, a little algebraic artifice 
can transform it into another which will converge. 

In the preceding chapter we have integrated in terms of cir- 
cular arcs. If we can also integrate the same expressions in 
algebraic terms, we shall have the numerical or algebraic mea- 
sure of circular arcs. Thus, much useful truth is revealed by 
two methods of integrating the same quantity, — and this is one 
feature of the utility of the science. 

For example, let us take the expression 

dx 



from equation (2), (Art. 67), whose integral is the arc of a cir- 
cle, the radius of which is unity, and sine x, and we shall have 
the numerical value of this arc. The above expression may be 
expanded as follows : 

L^=(l_a:2)-2^1+^a;»+i.|a;4^i.|.|a.6_j. &c. 



Jl'—x' 

Multiplying each term by dx, and integrating, we obtain 

u=zBm.-^x=x^——\-^-^-^l^^+ &c. 
2.3 ' 2.4.5 2.4.6.7 ' 

This integral requires no correction, for if we make x=0, u 

will become at the same time, as it should. 



INTEGRATION OF FRACTIONS. 267 

As X is the sine of an arc, it can never be taken greater than 
the radius, (unity,) and if we know the value of x for any par- 
ticular arc, that value substituted in the second member will 
give the linear measure of the arc. 

If we take «=30°, we know that the corresponding value of 
X is I, and taking ten terms of the series, we find 
Arc of 30°=0.62359877. 

Whence Arc of 180°=6(0.52359877)=3.14159262=rt. 

N. B. This problem is the same as example 1, (Art. 20), in 
the differential calculus. We repeat it here to develop the 
method of integration, and to show the harmony and beauty of 
science. 

For another example. One integration of the expression 

is the arc of a circle whose tangent is t, and radius unity. 

(See Eq. (5), (Art. 68.) 
For another integration we expand — —^ by division, which 
produces 

I__^2_j_^4_^6^jj8__^l0_|_ (fee. 

Multiplying each term by dt, and integrating, we have 

t^ t^ t"^ t^ (^ * 
Arc(tan.=0=^ — — + — — — + — — —4- &c. 
^ ^ 3 ^5 7^ 9 11 ' 

This result is the same as equation (4), (Art. 20), differential 
calculus, and therefore we will not again carry out the numerical 
result. 

For a third example. One integration of — is \og.x. We 

X 

cannot obtain another integration of this differential, because 
we cannot expand it into a series. 
• But if we place a:=l+y, then dx=dt/. 

And ^=J^. Whence r^= rJ^=log.(l+y). 
X 1+7/ ^ X ^ 1+y ^ ^ ^^' 

For the second integral we can expand into the series 

1— y+y'— y'+y'— y*+y'— y'+ &c. 



268 INTEGRAL CALCULUS. 

Multiplying each term by </y, and integrating, we obtain 

To determine the value of (7 we make y=0, then log.l=C. 
But log. 1=0, in all systems of logarithms ; hence (7=0 in this 
case. 

Had we made rf=a-}-y, the development of (1) would have 
been 

Now to determine the value of C we make 2/=0, then log.a= C, 
and the entire integral is 

log.(«4.y)=log.a+^— i^+ll— X &c. 

This result is the same in form as in (Art. 19), hence we omit 
carrying out the details, as it would be mere repetition. 
For a fourth example, 

Integrate — , or 



We perceive at once that one integral is log.(l-|-3a;^). That 
is, the hyperbolic logarithm of ("l-j-S^^) for the given differen- 
tial, is plainly the differential of the quantity divided by the quantity . 
We have got the transcendental mtegral, and now if we would 
obtain the algebraic or numerical integral, we must expand 
(\-\-ax^)~^, which is 

1 — ax^-^a'^x'^—a^x^'\-a^x^ — a^x^\ &g. 

Multiplying each term by 2axdx, and integrating, we have 

log. (l+«:i;2)=a:,2 _____[._______ ^g. +a 

which is the same as equation (1), example 3d, if we put y in 
place of ax^. 

(Art. 72.) All differentials in the form 

p 

x'^-^^dx(a-{'bx*)q 

can be integrated, term by term, aftei* expanding the binomial 
and multiplying each term by the part without the parenthesis. 



INTEGRATION OF FRACTIONS. 269 

But the series of integrals thus obtained, may not converge, 
for convergency will depend on x being less or greater than 
unity, and also on the signs of m and n ; hence the sum of tht' 
integrals, or the entire integral sought, may not be sufficiently 
near the truth to answer our purpose when obtained by such a 
process. Yet, when particular cases are given, algebraic arti- 
fices in the hands of a skilful operator, are equal to almost any 
emergency, and it is to such artifices we shall call the attention 
of the reader iu some future chapter. 



CHAPTER IV. 

Integ^ration of Rational Fractions. 

(Art. 73.) A rational fraction, numerically considered, is 
one which is less than unity, algebraically considered. It may 
be written in the form 

Fx''-\'q'x^^R'x^-\-S'x-{-T'" 

the highest power of the variable is greater hy unity in the denom- 
inator than in the numerator. If it were not, we would divide 
the numerator by the denominator, and thus obtain an integer 
term, and from the remainder and divisor, we would then form 
our rational fraction. 

Such fractions can be separated into a series oi partial frac- 
tions, whose denominators are binomials, provided the denomi- 
nator is capable of being separated into binomial factors. 

To separate a compound denominator into its simple factors, 
place the quantity equal to 0, and find the roots of the equation. 

Let a denominator be aj^+Pic^-i+^a;"-^ Gx'\-F; place it 

equal to 0, and let the m roots of the equation be represented by 

a, h, c, &c., then by the theory of equations, the denominator 

will be the product of {x — a), (x — 5), (x — c), &c. to m factors. 

These factors may be real or imaginary , equal or unegtcal. We 
18 



£70 INTEGRAL CALCULUS. 

shall commence with the most simple case, in which the factors 
of the denominator are real and unequal. 

Let the denominator of a rational fraction consist of the three 
factors (x-\-a), (x-\-b), {^-{-c); then the fraction will be equal to 
the sum of the three partial fractions, 
A , B , C 
x-\-a x-\-b x-\-c* 
the numerators A, B, and (7, are as yet undetermined constants. 

EXAMPLES. 

1. Suppose it were required to integrate the rational fraction 
(2x''—S)dx 

a;3 4/j. 

The denominator is obviously the product of the factors x, 
(x-\-2), (x — 2), therefore we may place the fraction equal to the 
three partial fractions, as follows : 

(2x^ — S)dx_ Adx . Bdx . Cdx . . 

" x^—4x ^"^^2~*"a:— 2 ' ^ ^ 

The integral required will be equal to the sum of the integrals 
of the three partial fractions. 

We can integrate the partial fractions after we determine the 
values of Ay By and (7, and these values are determined in the 
following manner : 

Divide ( 1 ) by dx, then we shall have 

2x''—S _A, B . 
x3 — 4x T x-\-2 x^' 
an algebraic equation, and nothing more. 

Reducing the second member to a comnion denominator, and 
^x^—S _( x^—4)A+(x^—2x)B+(x''+2x) C 
x^ — 4x x^ — 4x 

Omitting the common denominator, and transposing all to the 
first member, we have 

(2-^A—B^C)x^+2{B—C)x+{4A—3)=0. (2) 
As X represents a variable quantity, we are at liberty to make 
it equal zero. Or equation (2) will furnish the three equations 
(2— ^— J?— C>2=0, 2(5— (7>=0, 4^—3=0. 



INTEGRATION OF FRACTIONS. 271 

In short, by the theory of indeterminate coefficients, we have 
u4=f, 5=(7, and 2—^ -«_(7=o. Whence J5=|, and (7=|. 
These values put in (1), andinQW<,ting the integration, we have 

= 1 log.ar+| [log.(a:+2)+log.(ar— 2)]-^ C. 

_ \dx. 

Ans. ^log.(a;— 4)— ilog.(a;— 2)+C. 
Place x^—6x-\-S=0. Whence x=2, or 4. 

Therefore put „ = + , &c. &c. 

^ x^—6x+8 x—2^x—4 

(Art. 74.) The following example presents a case in which 
the denominator of the given fraction contains sets of equal 
factors. ^ 

3. Integrate ^^^ 



(a:— 1)2 (a;— 2)2 



Place - =-^— +^+ -^ -4-— ■ (1) 

{x—\y{x—<2Y {x—\y^x—\^{x-~2y^x—9, ^ ^ 

Clearing of denominators, and equating the coefficients of the 
like powers of x^ we have 

4^— 4^'+^--2^'=0. 
—4^+8^'— 25+5^'= 1 . 
A—5A+B—4B'=Q. 

From these equations we obtain A=l, A'=3, ^=2, and 
^'=—3. Whence 

/xdx r dx , o r d^ I Q r dx 

(a:— 1)2 (a;— 2)2 ^ {x~^Y ^ ^^"^ ^ (ar— 2)^ 

^/■^2=-^l+'^'^-^^'^~^2-'^^^-^"~'^+^- 



272 INTEGRAL CALCULUS. 

We integrate the first and third of these partial fractions by 
(Art. 61.) 

(Art. 76.) If the denon^''***^^^ ^^ * rational fraction contain 
imaginary roots, it T*>««t, contain a factor in the form 

ffmce this expression placed equal to zero, will give two ima- 
ginary values to Xy and since we know from the theory of algebra 
that imaginary roots necessarily exist in pairs, if there be m 
pairs of equal imaginary roots, there must be a factor in the 
denominator, in the form 

(aj^+Sar+a^+J^j". 

A rational fraction, as shown in the last article, can be sepa- 
rated into several partial fractions, and to the simple factor 

a;a+2aa;+a2+6S 
there will be a corresponding partial fraction 



which we propose to integrate. 

Put x-\-a=z, then dx=dz. And put ir—aJf= P. Then 

the fraction becomes -^^—dz, which is obviously the sum of 
two fractions. 
Whence f^^±^dz= r_^+ r_^. (1) 

The first term of the second member may be integrated thus : 

f /^=f l<'S-(^'+*^)- (Art. 64.) 

The second term is integrated by (Art. 70.) 
Whence fJ^^ =i!arc Aan. =^\ 

These values put in (1), give 

resuming the value of z in the second member. 



INTEGRATION OF FRACTIONS. 273 

It is proper to observe that an arc whose tangent is "^ , the 
sine of the same arc is — "^ — , and the cosine is 

These expressions afford the means of present- 

ing the proposed integral under different forms, designating the 
arc by its sine and cosine, in place of its tangent. 

EXAMPLES. 

i. Integrate 3 i~2 1 r 

The denominator is the product of the factors (1+a;) and 
(l-\-x^), therefore place 

x^—x-{-\ ^ A _, Mx-\-P 

Clearing of denominators, &c. we find 

^=f, M— — 1, and P= — ^. Whence 

^(x^ — x-\~\)dx ^ 3dx /- xdx r dx 

^ {JJ^x)(\+x^)~J 2(1+^ ^ 2{\+x^) ^ 9,{\+x^)' 
=1 log.( \+x)—\ log. ( l+a;2 )— 1 tan.- ' x. 

= log.(l4-a;)2— .log.{l4-a;2)*— .i.tan.-»a:. 
2. Integrate 1!^+?)^. 

X^ X"^' — 2x 

Arts. I log.(a;--2)+i^ log.(a;+l )— f log.ar. 

_- \dx. 

Arui, ^+|log.(ar+2)+^log.(a:-2)+a 



4. Integrate ^ 1 — ^-r-. 



Ans. •--ilog.a:+ilog.Va;2+a;4-2-l--4-tan.'^ /^?±i^+C7. 

2J7 \lJ7/ 



2J7 MV7 



274 INTEGRAL CALCULUS. 

By a review of the integration of rational fractions, we sliall 
perceive that the partial fractions to be integrated will fall under 
some one or more of the following forms : 

r_^, fx-dx, f—^^l , r-±-. 

(Art. 76.) We will now investigate a formula for integrating 

dx 
differentials in the form . > 

In the first place we will assume the equation 

/dx J[x ^^ J r- dx ^ ^ X 

in which K and L are indeterminate coefficients, and (1) will 

become a practical formula, provided we can determine K and 

L in terms of a and w^. 

To test this we must difierentiate equation (1), divide by dx, 

and clear the result of fractions, we shall then have 

\=K(x-'\oJ')—9.K{m—\)x^'\-L{x^^a'). (2) 

This equation must be true for all values of a;, it is true then 

when .T=0, and this supposition gives 

(^4.Z)a^ = l. (3) 

Equation (3) tal5;en from (2), and the remainder divided by 

x'^ , will produce 

3^4-i;— 2^w=0. (4) 

From (3) and (4) we obtain 

rr 1 . T 2m— 3 

-, and L: 



2(m— l)a2 2(m—l)a2 

dx 



These values of ^and L placed in (1), give C 



{x^^a^Y 



2m — 3 /> dx 



2(m— I)a2(a;2 4-a2)'»-i ' 2(m—l)a^^ (a^^-j-aay 
for the formula required. 



1. Inieffrate 



EXAMPLES. 

dx 



INTEGRATION OF FRACTIONS. 276 

Substituting 1 for a, and 3 for m, the first application is as 
follows : 

/dx X 13 r dx ,^ , 

For a second application of the same formula, we write 1 for 
a, and 2 for m, then 

r dx __ X ^ r dx /gx 

But J.^^=ta.n.-'x. (Art. 68.) (3) 

The result of (3) placed in (2), then that result placed in (1), 
and we have the final result as follows : 

r dx X . ^X ,3 _j , ^ 

2. Integrate 



(^^+6)^ 



r- dx X \ 5 f ^^ 



^ dx X . 3 ^ dx 



(a:2+6)s 24{x^+6y ' ^'^ (x^'+ey 

/dx X . ^ y- dx 

(a;2-j-6)2 ~ 'n(^^+6y'''^J x^-\'6 

_^=J-tan.--^. 



Whence C—^'^— - "^ -4-.___J^__ ,4- 

t^ (:r2_|_6)^ 36(.t2+6)3^24. 36(^:2+6)2 ^ 

. ^? I- ^ tan. -^-+a 

36.96(0:2+6)^36.96^6 " ^ 



276 INTEGRAL CALCULUS. 

CHAPTER V. 
Integration by Parts. 

(Art. 77.) In the differential calculus we have found thai 
the differential of a product, as (wv), gives the equation 

d(uv)=udv-\'Vdu. Whence uv=Cudv-\-Cvdu. 

Therefore fudv=uv — fvdu. ( 1 ) 

From this we perceive that the integral of udv can be found 
whenever we are able to integrate vdu. This method of inte- 
grating udv is called integration hy parts. 

The utility of this method of integration principally consists 
in its application to binomial differentials in the form 

x"'-'^dx{a-\-hx''y 
which are not integrable by direct methods. 

Many differentials in this form have already been integrated, 
but they were particular cases of this general form. 

In the following general investigation we may regard m and 
n and^, fractional or negative. 

In case p is a whole positive number, the binomial can be 
expanded, and each term can be integrated as before shown. 
When m and n are fractional, as they may be in particular ex- 
amples, as 

x^dx(a-\-hx^y 

place x=z^, z being a new variable with an exponent equal to 
the product of the two denominators. 

Whence x^dx{a^hx^)^=Qz'^dz{a-\-hz^Y 

Henc£y every binomial differential can he placed under the form 

Place x'^-^dx^dv, and (a-|-5a;") P =w. 

Then _=v, and du—bpnx''^^dx(a-\-bx'')P-^ 
m 



INTEGRATION BY PARTS. «t 

These values of u, du, v, and dvy substitued in ( 1 ) give 

Jx" - .rf^a4-fa-)P =f>±^£l!— ^*/^*"- <fc(o-|-&t').-' 

(2) 

Observe the identical equation 

Multiply the second member as indicated, and then we shall 
have 

Multiply each term by x'^-^dx, and write the sign of integra- 
tion, and it will stand thus : 

fx'^'^dxia+bx") P t=ajx'^'^dx(a+bx'')^^+ 
5j'ar'"+"-'(^a;(a+5ar")p-i. (3) 

If we multiply (3) by — , and add the product to (2), the 
m 

last term will be eliminated, and after a little reduction, the 

result will be 

Formula A. 

rx--^dx(X)p=^^i^^+ ^^^ rx-'^dx{xy\ . 

m-\-pn m-\-pn '^ 

in which X represents the binomial {a-\-b^). 

If we multiply formula A by (m-^jpn), change signs, trans- 
pose the first and last terms, and then divide by pna, we shall 
have 

F0RMIH.A B. 

Again, observing that the first members of (2) and (3) are 
identical, therefore the second members are equal. That is, 

afx^-'^dx(Xy^-^bJx^*^-^dx(X)P-^=^ 



278 INTEGRAL CALCULUS. 

Transposing and reducing, we obtain 
Formula C. 

Now if we transpose the first and third terms of this formula, 
and reduce the first member to unity, we shall have 

Formula D. 

^ ^ ' am am ^ ^ ^ 

The formulas (^4), (B), { C), (D), will apply to any possible 
binomial difi*erential that can be presented. 

When the exponent p is positive, and we wish to diminish it, 
we must use Formula A. 

When p is negative, and we wish to increase it, that is, dimin- 
ish it numerically/, we must use Formula B. 

When the exponent (m — 1) is positive, and we wish to dimin- 
ish it, we use Formula C. 

When that exponent is negative, and we wish to diminish it 
numerically, we use Formula D. 

The formula in (Art. 76) is substantially the same as for- 
mula B. 



(Art. 78.) It frequently happens that we are required to 
ials in 



integrate binomial differentials in the form 



Ja^'—x^ 
and for that purpose we can use formula C, and we now adjust 
that formula to this general case. 

For this purpose we must write in formula O 

a^ for a, —1 for 5, ~i for (i?--l). 

I for p, and 2 for n, 

then formula C will become 

Formula c. 



/ " — ii; Ja^ — X- ^ ••" ^ 



Ja^^x"" m+1 m+\J Ja^—, 



INTEGRATION BY PARTS. 279 

This will apply when (m-f-1) is positive, but when that expo 
nent is negative, we require the converse of this formula, which 
we find by transposing the first and last terms, changing signs 

and dividing by 

This will give 

Formula d. 



x^~^dx x^Ja^ x^ m-\-\ x^^'^dx 



We name this, formula d, because it can be drawn from the 
formula i>, the same as c was drawn from C. 

This formula must be applied when (m — 1) is negative. 
A formula corresponding to the particular form 
x^-^Mx 



Ja^+x^ 
can be deduced from (7, by substituting in that formula 
a^ for a, 1 for h, — ^ for {p — 1), \ for^, 
2 for n, and we shall have 

Formula C 



x^-^^dx x^Ja^ I x^ "^^* r ^""^^'^ 
J'^^J^ ^qii ~~m+lJ ^a2 _|_^2 



The converse of this is Formula d'. 

(Art. 79.) It is desirable to have a formula applicable to the 
binomial diflferential, in the form 

X^dx q_JL _i 

To integrate this by formula (7, we must place 
,7i_|_7i_l=g_x, n=\, h——\, ^— 1=— i, or jp=|. 
Therefore wp=|, and m=5' — |, and for a in the formula 
we must write 2a. 



280 INTEGRAL CALCULUS. 

These substitutions will change formula (7, into 

r_f!ri^^= — ^'^~^ ^ ^2a — X I ^aq-^a /> x'^~2dx 

Observe that ar**~^=-^, and «*i-ia:2=a;q-2, and-f!lJ=a;<J-~l. 
Jx Jx 

Therefore we can pass x^ under the binomial radical in- each 
term, and 

^ J^ax—x"^ q ~~ q J^ax—x^ 

To preserve uniformity of notation as much as possible, we 
will now write m in place of q^ and we have 

Formula d. 

r x'^dx __a;"'- 1 J^ax—x 2~ , g( 2?wr— 1 ) /- x'^'^dx 

"^ J2^'^^ m ^ J2ax—x^' 

This formula is to be used when m is positive. The converse 

of this is to be used when m is negative. To find the converse 

transpose the first and last terms, &c. and we have 

Formula d'. 

m r x^dx 



^ x'«'-^ dx ^ x^-ij2ax—x^ \ ^ f 



J%ax—x^ a(2m— 1) a(2m— 1^ J^ax—x^ 
Formulas d and d' diminish the numerical values of the expo- 
nent without the parenthesis, by unity. 

When m is a whole positive number, the final differential in 
formula d will be of the form 

/ ^- =ver. sin.- > --4-C7. (Art. 70.) 

J^ax — x^ ^ 

As we have before observed, the formulas A, B, C, and i>, 
are general, and some one of them will apply to any binomial 
differential that can be presented — but in consequence of the 
frequency of examples in which the sign of the square root ap- 
pears over the binomial factor, it is expedient to adopt special 
formulas, as c, c\ d, d\ to meet such cases. 



INTEGRATION BY PARTS. 



281 



We now give a few practical examples, which, together with 
the formulas, will sufficiently illustrate the whole subject. 



EXAMPLES. 



1 . Integrate 



',^dx 



Jl'-X' 



(Apply formula c.) 



In the first operation m+l=5, and a=l. In the second 
m-{-l=3, and so on. 






x^dx 



r xdx 



=-=-k^'Ji-x-'Hj 



xdx 



^1- 



-——Jl—x'. 



(1) 

(2) 
(3) 



To obtain the integral demanded, we must now take backward 
steps. That is, place the result obtained from (3) in (2), and 
then place that result in (1); and lastly add the arbitrary con- 
stant C, and we have 



the integral sought. 



2. Integrate 



x*dx 



J\—x'' 



(Formula c.) 



x*dx 



x'dx 



^ Vl=^ 4 ^^ Jl-^x' 



S 



x^dx 
dx 



=-f^AL=?l+i/- 



dx 



r ^ =sin.-> 



2 " J\—x- 

X. (Art. 67.) 



(») 
(2) 
(3) 



•282 INTEGRAL CALCULUS. 

Now the results obtained from (3) and (2) placed in (1) give 

N. B. When (m-\-l) is odd, as it is in the first example, the 
final integral will be dependent on the integration of ^ ^ - , 



or on -— a/1 — ^^• 

When (m-\-l) is even, as it is in the second example, the final 
integral will be sin.— ^a:. 

Hence, if (m-\-l) be a whole number, whether odd or even, 
the complete integration is possible. 

dx 
3. Integrate (Formula d.) 

In the first operation m — 1= — 3, m=^ — %. 
In the second operation m — 1= — 1, 77i=0. 

r dx ^_ yr=^ -{-i r d^ . (1) 

^ x^Jl—x^ 9,x^ ^ xj\—x^ 

r__i^_= VlE?-i f-^- (Formula fails.) 

Here we perceive that the formula fails in the second opera- 
tion, because m=0. Therefore we must find some other method 
of integrating 

dx 



xj\ — x"^ 

By an example in the differential calculus, (page 167), we 
learn that the differential of 

dx 



^iQg/ i+yi-^' ') 



xj\ — X 



Whence , 

xj\ X 



^.-:^=-'°K'±#^) <" 



* This example was designed for page 167, but was omitted by mistake. 
We shall now place it among the miscellaneous examples. 



INTEGRATION BY PARTS. £83 

And this value placed in (1) produces 



)dc 



(Art. 80.) The method of finding the integral of 

dx 



xj\—x^ 

by mere reference to the differential calculus, is not satisfactory 
to a learner. It is therefore desirable to obtain the integral 
directly, as in other cases. 



To this end assume ^1 — x^=Zy xdx= — zdz. 

Whence f— ^-= r— ^= r^_=^^^^. 
^ xj\^^ ^ 1—2' ^ 2'— 1 2—1 2+1' 

/• dx , p dz ,y r dz 

=:~il0g.(^-l)+il0g.(g+l). 
' ^ 2—1 

N. B. The product of two factors is the same when the 
signs of both factors are changed. Thus -\-P multiplied into 
— Q produces — PQ, Also, — P into -{-Q, is — PQ. 

Therefore we may change the signs of each factor in the 
second member of the equation above. Then we have 

f '^ -=-;-iog. l±£=-iiog.i+^.i±?. 



=-iiog.i±i 



=— ilog.i+f_. 

=-iog.(±b/i^) 



284 INTEGRAL CALCULUS. 

4. Integrate — (Formulae?.) 

Here m=2 in the first operation, and unity in the second ope- 
ration. 

r a;^ dx __xj^ax-—x^ \?^ r___^___. (1) 

j__ xdx _ ^_^-^^^^z:^^^j dx ^2j 

J2ax—x^ ^ Jtax — x^ 

r ^"^ - =ver.sin.-^l (Art. 70.) (3) 

JSLax—x^ « 

Whence, by substitution, we have 
r ^'^^— =-/^y+gg:^ V2^^=i^+?^ver.sin.-»^+a 

6. Integrate ?I_. (Formula i>.) 

In which ^=0, m — 1= — 3, w»= — 2, «=1, in the first 
operation, 

6. Integrate ^ . (Formula C.) 

fw+l=2, a=a», 6=1. 

r_4!^=^V«+^^— ^ r ^^^ (1) 

Ja-\-bx 3 ^ Ja'\-bx 

J ^dx _ ^ aJ^j^x _ J^'^og'Ja+^ ^ (Art. 62.) (2) 
Ja-\-bx b^ ^^ 

Place a-|-^a:=s, and integrate by an independent process. 
Whence by substitution we shall have 



INTEGRATION BY PARTS. 285 

7. Integrate -^!^. (Formula 0.) 

a-{-bx 

-^ a+bx 36 2b^^ b"* b^ 6 v "i^ /"r 

8. Integrate ^^^ -— x. (Formula D.) 

One operation gives 

^ _c^ 1_ __7ft r_J^_ 



9. Integrate dxja^'\-x'^s (Formula^.) 

Here m — 1=0, P='h »==2, a=ia'. 

~i . a^ r dx 



J dxJa^-\rx^==''_J^^+~J- 



2 ^ Ja^+x' 

In the diflferential calculus we are taught by an example, 
(page 167,) that the differential of log.(a;-|-<ya^+a:*) is 

dx 



Ja^+x'^ 



dx 



Conversely then f ^"^ =\og,{x+Ja^+x^). 



Therefore Jdx^a^ +x^ ^ xja^ + ^^4-— log.(a;+ ^g^ +a;« ). 

2 2 



10. Integrate dxjx^ — a!" 



Ans, x jx^—a^ _ g_log.(a;4-Va;^— «^ )■ 



(Art. 81.) The last two problems require us to integrate 

differentials in the form ^ independently of the for- 

Jx^±za^ 
mulas in (Art. 77), and to infer the integral, as we have just 
done is not satisfactory, therefore we operate as follows : 

The square root of x^zta'^ obviously must contain ifca?, and 
19 



286 INTEGRAL CALCULUS. 

some other quantity which we can represent by z. Therefore it 
is natural to place 

(1) 





Ja^ 


-\-x^ =.Z- — X. 




o2 = 


=-^2xz+zK 




c^- 


~x)dz=izdx. 


dz_ 


dx 


_ dx 



Ja^+x' 



dx 



^^""^^ /-^=log.^=log.(a:+V«^+^^). 

10. Integrate ^^^^^^ (Formiila J?.) 

N. B. This example, as well as several others, will be found 
in the differential calculus, in the first part of this volume. 

Here, m — 1=% — 1, or m:=n, but w in the formula referred 
to this example is 1, and a=l, p — 1= — n — 1, or p= — m. 

/^»-w;.(i+;.)-»-i==£li±5):!-i!!=^/-fl:i^. 

Tl fir I l—r'Xi 

But the last term of this equation is zero, because (n — n) is 
zero, whence 

== — , the integral sought. 

(l+x)^+^ (l+xf ^ ^ 



1 1 . IrUeffrate _ dx(l+Jl—x^ ) 

x^J\—x^ 
This can be separated into two parts. 
— dx dx 



Thus 



x^J\—x^ «^ 



The integral of the first part is J^ — ^ ^, and of the second 

x 

it is — , whence the whole integral is ^"Hn/ ^ — ^^ . which is equal 
^^ X 

to V^+^+yi— a; ^ The differential of this last quantity was 

^1+^ — J^ — X 
demanded in the^ differential calculus. 



I 



INTEGRATION OF FRACTIONS. 287 

CHAPTER VI. 

Integ:ration of Irrational Fractions. 

In the last chapter it was ifound that differentials in the form 
cannot be integrated by {)arts, unless we can integrate 



Ja+bx' 

rential fraction in the form 

^a+bx' 



the differential fraction in the form - . which may be an 



irrational fraction. 

(Art. 82.) The object of this chapter is to develop the gen- 
eral theory of integrating differentials in the form 

^^ and in the form -"^ 



JA+BX+ Cx^ JA+Bx— Cx^ 

Our first object is to find equivalent expressions in which x^ 
shall stand loithout a coefficient, and with the plus sign. In other 
words, the coefficient of x^ must be 4-1' ^"^ the first case it is 
obvious that 

, dx , 

dx - dx 

JA+Bi+G^" JcJIA+^x+x' ""V^VH-^^T^" 

^ ^ (1) 

If ^=a, J?=5. Orif^=aC, B=bC. 

In the second case 

dx dx 

jA+Bx-^Cx^'~'j^'^J-^+-^x+x^ '^ J~CJa+bx+^ 
^-C ^G (2^ 

If ~4=«' -4=^- O^^f ^==-a(7, B==-bC. 
G G 



268 INTEGRAL CALCULUS. 

By inspecting ( 1 ) and (2) we perceive that the integral in 
each case will depend on the integration of 

dx 



which integral must be multiplied by — ^rr-* ^ov examples in the 

JG 

first case, and by , for examples in the second case. 

■J-o 

Henee our exclusive attention; will be directed to the integra- 
tion of 

dx 



Ja-\^X'\-x^ 

the result of which we shall multiply by — -i~ for a general 

vdbC7 
formula. 



Place Ja-\-bx-^x^ =5? — x, * ( 1 ) 

Squaring and reducing in part, we have 

Whence (h-\-2z)dx=^{z^x)dz. {%) 

Dividing (2) by ( 1 ), and 

(5+2. j^^^^ (3) 

Therefore 
C ^^ = r 2( fe _^ 2 p dz r-» 

"^ jlfjl^+b^:^''^ Jc(b+2z) Jo^^H^^' 

Again, place 6+2s=^. Then dz= — 

*It is more natural to place the radical equal to z-\-x, but as both — x 
and -f-ar will give a: 2, we can take either, and the minus sign will give a 
more convenient result than the plus sign would do. But in numerical 
examples we may take either one. 



INTEGRATION OF FRACTIONS. 889 



But in ( I ) we find z=x-\- J a-\-bx-\-x^ . 
Whence /— ^ =^\og.{b+2x+2ja+bx+i^. ) 
Finally, 
f "^"^ = ^ log.(&+2a:+2Va-f^a?4^M-f const. 

EXAMPLES. 
J' 

1. Integrate — . (Formula J*.) 

Jl+x- 

Here (7=1, ^=0, and ^=1. Therefore a=l, 6=0. 



cfo: 



Whence / — 'rt-=\og.{2x+2jl+x^)+c. 



But \og.(2x^2jl+x' )=log.(a;+7l+«^ ) +log. 2, and log. 
2 may be united to c, and become part of the arbitrary constant. 



dx 



Therefore J ^-^ =\og.{x+^l+x'' )-fc. 
Jl-\-x^ 

N. B. In some of the following examples the results of the 
formulas may be reduced by expunging the factor (log.2) and 
conceiving it to be added to, and to become a part of the arbi- 
trary constant. 

2. /_,£_=log.(ar+V^^T)+c. 
Jx^—1 



dx 



Jx-^-x^ 
4. / l^-^ =log,( 1+2.^4-2 V l+^^j+c. 

f. dx 1 



J^log.(a:+V;c»— l)+log.2-f-c. 



290 INTEGRAL CALCULUS. 

N. B. Formulas in other works give 

for the integral of the second example ; both are correct ; indeed 
they are equal, as one can be reduced to the other as follows : 

log/ _^+jJ^^_ZZl) is equal to log.(a?+^a?2— l). 

If we multiply numerator and denominator by ^ — 1, we shall 
have 

iog.M:JEEn= iog/^>/=lt^5El! V 
1 \ J— I / 

log.(;r ^=1+ VT:=? )— log. v=T. 
Whence 

The last expression is applicable to circular arcs, as will soon 
appear. 

6. / ^^ ^Xlog,(x+h+xn+c. 
^ Ja+bx^ Jb ^b 

But -Liog.(a.V6+7^q:^^)=.iiog. (+Jl-\-x^)+2jog,b. 

4h Jh ^* Jb 

and either of these expressions differentiated, will produce the 
given differential. 

(Art. 83.) In treating of circular arcs in the differential cal- 
culus, we have found that when the radius of a circle is unity, 
and the sine of the arc is x, the cosine of the same arc must be 
Jl — x"^, and the differential of sine is 

dx 



and this was the quantity to be integrated in example 5. There- 
fore another integral of that quantity is the arc of the circle 
corresponding to the sine x plusy an arbitrary constant. 
Let z be that arc, then a;=sin. z, J\ — a;2=cos.2. 



INTfiGRATION OF FRACTIONS. 29 

And /_^=.4-C. (1) 

But by example 5, we have 



dx 



S-^^==-L=^og.(zJ-l+Jl-x')+c. (2) 

Ji-^' V— 1 

Whence z+C=—^\og.(xJ^^+JT^)-^c'. 

To determine the relation between the constants, we will con- 
ceive the arc and its sign to commence at the same point and 
increase together, and suppose x=0, then will z=0, and the last 
equation will become C=c', that is, the constants will be equal 
to each other, and therefore they may be omitted and tjie equa- 
tion itself will become 



V-1 



Substituting the values of x and of ^1 — x^ in this last equa- 
tion, and we have 

log,(sin.0^ — l-|-cos.0) = ^ — l.s. 

Multiply each member by log.e, observing that e is the base of 
the hyperbolic logarithms, and its log. is 1, and 1 as a factor 
maj/ be made visible or invisible. We will make it visible in the 
second member, then 

log.(sin.3^ — l-\-cos.z) = ^—l,z\og.e. 
=log.(ev^— ^-2). 

We can now omit the sign (log.) in each member, which is in 
fact passing to the numbers ; then we shall have 



sin.^^ — l-|-cos..<2=6v/— i.z. (1) 

If we take z negative, we shall have 

sin.( — z)= — sin.2;, cos.( — z)=s(iOS.z. 

And the final result will be 

-— sin.0^^-|-cos.^.=.e-vA^i-^. (2) 



292 INTEGRAL CALCULUS. 

By adding (1) and (2), and dividing by 2, we have 

C08.g= ^^^'+g~^^' . (3) 

2 

Subtracting (2) from (1)> and dividing, we obtain 

Bm,Z= -::;:^ . /4) 

2V— 1 

By substituting n^ for ^ in ( 1 ), we have 

Qm.nzJ — \-\'CQ&.nz=^e^—^'^. (5) 

Going back to ( 1 ) and raising each member to the wth power, 
we have 

{sin.07^+cos.g)"=ev^-i°^ (6) 

The second members of (5) and (6) are identical, therefore 

(sm.zj — l-[-cos.g)°=sin.Mg^ — l-\-cos.nz. (7) 
These expressions are purely algebraic symbols, expressing 
the relations between the arc and its sine and cosine, which, by 
proper artifices can be developed in numerical quantities. 

Equation (7) is the same as appears in Robinson^s Geometry, 
page 223, and its practical importance and utility is there shown. 

(Art. 84.) In (Art. 71) the diflferential is integra- 

ted, and the result is a numerical series. But the integral found 
is a logarithmic expression. The two integrals deduced from 
the same differential must be equal to each other. That is, 

This is true for all values of x. Then by supposing x=0, we 
find (7= C", the two arbitrary constants equal to each other. 



Sx^ 



Hence, -i- W.(a;+ Jar^— 1 )= x+—-+ "^TL^+Ssc. 
jZZ\ ^ ^ ^^ ^ ^2.3^2.4.5^ 

But this is an impractical equation on account of the presence 
of the imaginary factor. 

Examples 2, 3, and 4, (Art. 82,) can be expanded into series 
and integrated term by term. 



INTEGRATION OF FRACTIONS. 293 



Then we can have the numerical values of \og.(x-\-Jx^ — 1)» 
and onog.(l-{-2x-{-2jx+x^), and of log.( l J^2x-\-2jl -\-x-\-'x^) , 
but it is not important to obtain them, because we have already 
found a simple and general logarithmic series in Chapter III, 
(Art. 71.) 

(Art. 85.) We can find another integral to the differential 
by another method of integration, as follows : 



Ja-{-bx — x^ 

If we place a-\-hx — x^=0, and resolve the quadratic, we 
shall find two real roots. Let them be represented by r and /. 

Then x^ — Ix — a=(ar — r){x — r') 

Changing signs 

a-\-bx — x^ = — {x — r){x — r')=(ar^— ?•)(/ — x) . 

This being understood, we can assume 



Ja'\-bx—x'^ = J{x—r){r—x)=(x—r )t. (1) 
By squaring, and afterwards dividing by {x — r), we obtain 

r'--x={x-^')t^. (2) 

Taking the differential, and 

—dx=i'^dx+'ltdt{x—r ). 

Whence cf^=_?Mf::^). (3) 

1+^2 ^ 

Dividing (3) by (1) will give 

dx 2dt 

Jc^hx—x^ l+«^'' 

Whence J ^^ ==(7— 2tan. -*(<). (Art. 68.) 

Ja'\-bx — x^ 

= (7— 2 tan.- » 1^'—^ 
'^x-r' 

the value of / taken from (2). 



294 INTEGRAL CALCULUS. 

CHAPTER VII. 
Integration of HSLponential Differentials. 

THE SERIES OF JOHN BERNOULLI. 

(Art. 86.) A differential in the form Xa^dx can easily be 
integrated, provided Xbe an algebraic function of a;, and in such 
a form that successive differentials will terminate in a constant. 

To establish a formula to integrate Xa^dx, let us call to mind 
the well known equation 

jFdQ=FQ-jQdF. (1) 

Now let F=X, and dQ=a^dx. (2) 

To integrate (2) we will put y==a^ , whence log.y=a?log.a, 

And dy=\o2.a.a^dx, or fa^dx=-l—=: 

^ ^ ^ log.a log.a 

That is, e=T^. (3) 

log.a 

Again, assume 
dF==dX=Xdx, dX'=X"dx, dX"=X"'dx, &c. (4) 

Here we perceive that X', X'\ X"\ <fec. are the successive 
differential coefficients of X. 

The values of F, Q, dF, dQ, taken from (2), (3), and (4), 
and substituted in (1), give 

fXa-dx^:^—-^ CX'a^dx. (5) 

•^ log.a log.a*^ ^ ^ 

Again fXa^dx=?^^-^ rX'a^dx. (6) 

^ log.a log.a^ ^ ' 

And CX"a^dx=-^l^—l— CX"'a^dx. (7) 

^ log.a log.a^ ^ 

&c. &c. &c. 

If we substitute the values found in (7), in equation (6), and 

then that result in (5), equation (5) will become 

/•Z«>d^=jgl-/'"' + -^"°' - ' CX"a^dx. (8) 
-' log.a (log.a)» (log.a)3 (log.a)^'^ 



EXPONENTIAL DIFFERENTIALS. 296 

If X" is a constant quantity, dX"=:X"'dx==0, and the series 
terminates with the third term, and in general the series will 
terminate with the term in which the last dififerential coefficient 
becomes constant. 

If a becomes e, the base of the Naperian system, then log.a 
becomes log.e=l, and the formula preceding becomes 

JXe^dx=e^{X-^X'+X"—X'"+ &o. &c.) (9) 

EXAMPLES. 

1. Integrate e^x^dx. Ans. e^(x^ — Sx^-{-6x — 6). 

In this example X=x^. Hence X'=3x^\ X'=6x, 
X'"=6, and X""=0. 

2. Integrate e^x^dx. Ans. e^{x^ — 2x-\'2). 

3 . Integrate e^(x^ — f )^dx. 

Ans. e''(x'^^4x^-\-9x^ — 18ar-f20i.) 

4. Integrate e^(x^-\-Sx^—l)dx. Ans. e'^(a;3— 1). 

Remark. — We can extract the cube root of any number which 
is a perfect cube, with comparative ease, but when the root is a 
surd, we can only approximate to it by a series. So it is with a 
differential. When an exact integral exists, we can find it with 
comparative ease, but when no exact integral does exist, the ap- 
proximate integral can be obtained only by a series. 

In the Mathematical Operations, page 321, we required the 

differential of , and found it to be ( ' , — )e^dx. Con- 

sequently the integral of this last expression is , and it is 

probable we can extract it from the differential as a particular 
case — but we could not be sure of integrating any other exam- 
ple of a similar form. 

In this example X=A"t"^^^ , hence X', X", &c. do not 

converge toward a constant, and therefore it will be useless to 
apply the last formula. 



296 INTEGRAL CALCULUS. 

The solution of sucli examples will depend much on the skill 
of the operator, guided by general principles. 

dx 

-X 

e^dx , r^^^dx 



Whence C-l — ^^-j-J^ff — ?=the required result. (1) 



Let P=_^,and dQ=e''dx. Whence §=c^. 
1 — X 

Substitute these values in equation (1), (Art. 86,) and we 
have 

/xe^dx B^ __ ^ e^dx 

Transpose the last term, and 

/e^dx , rxe^dx e^ ,-,* 

The first members of (1) and (2) are identical, therefore 

The required result = 

In the same manner integrate the following differential : 

e^dx(--\-\og.x\ Ans. e^\og,x. 

(Art. 87.) When X——^, the successive differential coeffi. 

X 

cients of Xwill not approach a constant, and consequently for- 
mula (9) in such cases will be of no practical value, and we 
must return to first principles, and seek the integration of 

a^dx 

We will apply the principle of integrating by parts according 
to the fundamental formula, 

JPdQ==PQ^jQdr. (1) 

Here P=--, and dQ=a^dx. Whence ^= -^. 
X log.a 



EXPONENTIAL DIFFERENTIALS. 297 

Substituting the values of P, Q, dP, and dQ, in (1), we 
obtain 

/^a'^dx^ a* 1 n ^ a^dx 
"^ '1^ a;"log.a log.a*^ x"^^' 

Transposing the first and third terms, dividing by and 

changing signs, we shall have 

/a^dx a^ I log. a ra^dx 

Now if we write n for w+1, we must write n — 1 for n, then 
the preceding formula will become 

a formula which produces a continual diminution of the exponent 
n. When n becomes 1, the formula fails, for then the factor 
(w — 1) in the second member, becomes 0. 

Hence the differential ^ — - must be integrated approximately 
x 

by a series, no finite integral corresponding to it has been found, 
for the very probable reason that none exists. 

By Maclaurin's theorem we expanded a^ (Art. 18,) into the 

series l-j-f?-4-f 4-_£J^ — , &c. Multiplying each term by — 

^1^ 1.2^1.2.3 ^^ ^ X 

and integrating, we shall have a converging series when x and c 

are each less than 1, or when the product ex is less than L 

When the exponent w is a fraction, it will also be necessary to 
complete, or rather approximate to the integral by a series. 

The following examples will illustrate and show the method of 
integrating exponential and logarithmic functions more clearly 
than anything else, for no general formulas can meet every case 
and condition. 

EXAMPLES. 

1 . Integrate — 



298 INTEGRAL CALCULUS. 

Place ^=a^. Then ^a;=— ^_, and s"=a'". 

zloof.a 



Whence 



^ a^dx ^ 1 r dz 



which is easily expanded into a series, and then we can integrate 
term by term. 

2. Integrate x"(log.a;)°ffa;. 

Place z^=\og.Xy x'^dx^^dPy and P=(log.a:)"5=g'', and 
integrating by parts, we have 

Cx^(\og,xfdx= ^""^ '(^Qg-^)!-, _^_ Cx^aog^xy-^dx, 

Substituting for n successively a;— 1, n — 2, &c. we shall find 
J'a;"'(log.a;)"£/a;= 

This series will terminate whenever ?i is a whole positive 

number. 

This series fails when m= — 1, for then m+l=0, which woiUd 

x^-^'^ x° . /. . 

make the factor ^ = — , or infinite. 

m-j-l 

But when m= — 1 the differential becomes 

(\og.xYdx 

X 

dx . 

and this is very easily integrated, for — is the differential of the 

log.fl?. Therefore 

dx 
Place 2=log.a;. Then c?2= — , and the differential becomes 

X 

^dz. Whence fz^dz=.^=(}S.^-. 
This is subject to the exception n= — 1 . 

3. Integrate log.xdx, by parts, Ans. ar(log.a; — 1). 

4. Integrate -~^—.. Ans. . 

xiog.^x \og.x 



EXPONENTIALS INTEGRATED. 299 

5. Integrate ^^ \og.x. Integral ^-^--{-Xo^^Al—x) 

(i—xy ^ \—x^ *" ^ ^ 

dx / 1 \ 

6. Integrate '. — _ log. ( ). 

xjx VI— a;/ 

Integral Slog. ^+V^ -g-w/^-\ 
l^J^ Jx ^\\-xJ 



dx 

xJx 

Integrating by parts will give us 



Place dQ=^-!^, and P=log/_L\ 



^-Llog._L+2r J^ . 
Jx 1—^ ■" Jx{\—x) 

(Art. 88.) The method of integrating by parts produces 
THE SERIES OF JOHN BERNOULLI. 

This series is remarkable for its similarity to the series of 
Taylor, and it applies to the integration of quantities in the form 
XdXy in which X is any function of x. The process is as follows : 
JXdxz=Xx-^JxdX. (1) (By parts, Art. 86.) 

But xdX=- xdx. Integrating this last expression by parts, 

dx 

conceiving =i', and xdx=dQ, we shall have 

dx 

^ *^ dx dx 2 *-^ 2 dx ^ 

Again, 

"^ 2' dx ^ ~dx^' 2 2.3 dx^ ^ 2.3 dx^ ^ 

Substituting in succession these values in (1), that equation 
will become 

fXdx=Xx^^,^-+^.J^-^ &c. +C, 
^ dx 1.2 ' dx"^ 1.2.3 

the sei'ies in question. 



300 INTEGRAL CALCULUS. 

To illustrate this series, let it be used to integrate x^dx, al- 
though in practice it never should be applied to such examples, 
because the integration is too simple to require it. 

Hence X=xK Whence ^=3a;». ^1^=2, S.x, and 

dx dx^ 

^'^ :2.3. 



dX^ 
Therefore by the series 

^ 1.2 1.2.3 1.2.3.4^ 

The sum of this series is j — +(7 Y the true integral by the 

common method of integration. 

The utility of this formula will be apparent in the following 
example, which is new to us, and it shows the beauty of analysis 
as clearly as any thing we ever met. 

In the differential calculus, (Art. 18,) we find the following 
expressions : 

c?.sin.ar=cos.a;f^. (1) 

(/.cos.a?=— sin.ic dx. (2) 

Whence sm.x= f cos. xdX'\'G. 

To apply the series of John Bernoulli, we must make Jr=cos.ir. 
Then, by successive differentiation, we have 

dX . d^X ^ d^X . d*X . 

: = — sm.rc, ==— cos.rr, =sm.ar, ^=cos.a?, &c. 

dx dx^ dx^ dx'^ 



r* 1 I sm.aj.ic cos.ar.a;*' sm.ar.a?^ , 

sin.a;= / cos.* dx^QOS.x,x-\- - — .<> ___ oiu..<^. ^^. 



_c^£^ __sin£^_ ^^ ^^ 
1.2.3.4.5 1.2.3.4.5.6 

It is not necessary to add the constant (7, for if we make 
a?=0, sin .a? will equal 0, and each term of the second member 
will equal at the same time, hence (7=0. 



EXPONENTIALS INTEGRATED. 301 

Now divide every term of the last equation by cos.a.', and for 

— — write its equal, tan.iP, and we shall have 

COS. a; 



tan.a;=a?-|-tan.a?. — — tan..i;- 



1.2 1.2.3 1.2.3.4 ' 1.2.3.4.5 



tan.a; '- — &c. 

1.2.3.4.6.6 



Uniting the coefficients of the tan.a?, and we have 

/^l— ^'— 1-_ 1' — - + &c.Van.a: 

V 1.2 1.2.3.4 1.2.3.4.5.6 ' / 

V 1.2.3^1.2.3.4.5 1.2.3.4.5.6.7 / 

X + + &c. 

1.2.3^1.2.3.4.5 1.2.3.4.5.6.7' 

Or tan.a^= -^ -^ ~ (A) 

1— ^-+ - — - -I- &c. 

1.2^1.2.3.4 1.2.3.4.5.6 

But tan.ar=-^. (B) 

cos.a; 

Equations {A) and {B) are but different fornis of expression 
for the tangent a;, and as the fractions are irreducible, we may 
conclude at once that 

X^ X^ X* 

1.2.3' 1.2.3.4.5 1.2.3.4.5.6.7^ 

And cos.ar=1 — -f- — ? — ^ 4- &c. 

1.2 ' 1.2.3.4 1.2.3.4.5.6 ' 

These useful and beautiful formulas were found in our appen- 
dix to trigonometry, but the process there is much more complex 
than this one. They were also found in the differential calculus, 
(Art. 18.) 

(Art. 89.) We can use this series to integrate a logarithmic 
differential. 

For example, the differential of log.(a+a:) is -— . There- 

a-^rX 

20 



302 INTEGRAL CALCULUS. 

fore log.(a-|-^)= f > but this second member can be inte- 

grated by the series of Bernoulli, f —. 

^ a-\-x 

Here ir=_i-. Whence ^=- ^ 



a-\-x dx {^a-\-xY 

d^X__ 2 (^«Jr___ 2.3 d^X_ 2.3.4 

<^a;2 (a+«)^' c?a;3 (a+rc)*"' ~dx^ {a+xy 

mdx mx , mx^ , mx^ , 



&c. 



log.(«+^)=/ 



&C.+C7. 



This is true for all values of x, it is true then when x=:0, and 
making this supposition, we have log.«= C. Whence 

log.(a+.)=log.«+»(-^+_^_+ 



&c 



.) (.. 



3(a-\'x)^ ' 4{a+x) 

If we assume m=l, this equation will correspond to the A^a- 
perian system of logarithms, and if we assume x=\, the equation 
will become a very simple and practical formula for computing 
logarithms in that system. 

It will then be the following : 

log.(a+l)=log.a4-/^_i-+ J _ + 



^ ' ^ ' 1 &c. 



) (2) 



3(a+l)3 4(a+l)^ ' 5(a+l) ^ 

In practice we may apply either (1) or (2), as we please; (1) 
has more scope than (2), because x can be any number, whole 
or fractional. By (2) we can find the log. of (a+1 j when the 
logarithm of a is known. 

Either of these formulas can be used for computing common 
logarithms when m becomes known. 

The value of m is discovered for the common system by com- 
paring the log. of 10 in each system. (See Algebra, p. 241.) 



EXPONENTIALS INTEGRATED. 303 

To make a table of logarithms corresponding to any system, 
we are compelled to commence with the Naperian system. We 
must continue in that system until we obtain the Naperian log. 
of 10. Then we can find m, and then we can pass to the com- 
mon system. 

We have explained this whole subject several times before — 
but its great utility and beautiful philosophy is a sufficient excuse 
for a repetition in connection with this new formula. 

This new series does not converge as rapidly as some others, 
but it is more symmetrical, and was obtained by fewer steps than 
any other. 

To find the Naperian logarithm of 2, we must make «= 1 in 

equation (2), then log.a=0, and =i. Hence we may write 

the series 

l-l—J 4- L_4-_J__4. L_, &c. &c. 

2 ' 2(2)^ ' 3(2)3 ' 4(2)4 ' 5(2)5 

Now if we take |, or . 6, and divide it by 2 continuously, we 
shall have 

2^(2)2^(2)3^(2)* 

The first term of this series divided by 1, the second by 2, 
the third by 3, &c. &c., and the sum of these will be the loga- 
rithm sought. The work would stand thus : 

1).5 5000000 

2). 25 1250000 

3).125. . . • 0416666 

4).0625 0156250 

5). 03125 0062500 

6).015625. 0026041 

7).0078125 0011161 

8). 00390625 0004882 

9).001953125 .0002170 

10).0009765625 0000976 

&c. &c. 

'^6930636^ 



304 INTEGRAL CALCULUS. 

This operation continued and extended to a greater number of 
decimals, would give the true hyperbolic log. of 2. The fore- 
going is only designed to show the practical form of making the 
computation. 

If we multiply the log. of 2 by 3, we shall have the log. of 8 
at once. Then make a=8, and again apply the formula, and we 
shall find the log, of 9, which divided by 2 will give the log. of 
3, &c. &c. 

(Art. 90.) If the object were simply to obtain the best prac- 
tical formula for computing logarithms, we would integrate the 

differential — by two different methods. First, by rational 

fractions, (Art. 73); second, by expanding _ into a series 

a* — x^ 

by division, and multiplying each term by dx, and integrating. 

Then we should have 

^ fl2 — ^2 \^^ — y.y 

f ^^^ =f-f-— 4-_fl-^£l.-U&c. 4-C'. 

Whence log. /^^±?V 2/^- +—+—+—+ <fc<^-H (1) 

Equating the second members, and making a;=0, there will 
result C=C'i and no constant appears in (1). 

If we make ^±?=1 4-1, then will ^=-_L-- , and (1) will 
a—x ^z a 20+1 ^ ^ 

become 

log/l +l^=log.(^-|-l )- log.0= 

2/^-_J— + ^ I- \ 4- <fec.\ (2) 

V^204.1~S(22+l)3^6(20+l)5^ / ^ 

the most approved formula yet found. 

If we make a=2, and ir=l, in formula (1) we shall have the 
Naperian log. of 3 at once. 



MULTIPLE ARCS. 305 

The operation is as follows : 

.5 600000000 

3). 125 041666666 

5).03125 625 

7) 78125 1 1 16071 

9) 1953125 217013 

1 1 ) 0488281 25 44390 

13) 1220703125 9390 

15) 30517578125 2033 

17) 7629394 439 

.549306002 



1.098612004 



We commenced the operation with .6, divided this by 4, pro- 
ducing the next line below .125. This we again divided by 4, 
thus finding the next line below, and so on. These sums we 
again divided by 1, 3, 5, 7, and so on, producing the next col- 
umn. The result is true as far as six places of decimals — the 
seventh decimal should be 3 in place of 0. 

The double of this logarithm will be the hyperbolic log. of 9. 
Then we might make z=9, and formula (2) would give the 
hyperbolic log. of 10. 



CHAPTER VIII. 



The Integration of Circular Differentials of 
multiple Arcs. 

(Art. 91.) The object of this chapter is to investigate and 
show the method of integrating dififerentials in the form 
sin.'^xdx, siu.^xdx, or in general, sm.''xdx. 

For a clear comprehension of this, we must re-examine the 
method of taking the dififerentials of functions in the form 

sin. 3a;, or sin.nx. 



306 INTEGRAL CALCULUS. 

Let 2=w.r. Then dz=ndXf sin.0=sin.war, d. sin. z= cos, sdz. 
That is, d. sin.nx =n COS. nxdx, and d.G08.nx= — nsin.nxdx. 

Whence rcos.nxdx=-^-^'^, and fsin.nxdx^—^'^t^JA) 

These primitive formulas will serve as our general rules of 
integration in this chapter. 

As a preliminary step, we require formulas for sin.^a;, sin.^or, 
sin/ir, or in short, sin. "a; expressed in a series of the simple 
dimensions of the sines or cosines of multiples of that arc. 

Let y and x be any two arcs, then by trigonometry we have 
sin.(y-|-a;)= sin.2/cos.a;-j-cos.2/ sin.a;. 
sin.(y — x)=^sm.y cos.x — cos.ysin.a:. 

By adding these two equations and transposing s\n.[y — x) we 
obtain 

sin.(y-|-a;)=2sin.ycos.a; — ^sin.(y — x). (1) 
Now suppose y=nx. Then (1) becomes 

sin.(w-{-l)a;=2sin.wa; cos.aj — sin.(% — \)x. (2) 
Making n=\, 2, 3, 4, 5, (fee. in succession, we form the fol- 
lowing table : 

sin.2«=2sin.a;cos.a; — 0. (3) 

sin.3aj=3sin.ar — 4sin.*a:. (4) 

sin.4.i'=(4sin.a; — 8sin.2a;)cos.ar. (5) 

sin.5:»=6sin.a; — 20sin.3ar+16sin.5.r. (6) 
(fee. (fee. 

Again, if we take the trigonometrical equations 
cos.(2/-|-a;)=cos.y cos.a; — sin.?/sin.iP. 
cos.(y — ^a;)=cos.y cos.ir-|-sin-2/sin..r. 
Add them and transpose as before, we shall have 

cos.(y+a;)=2cos.2/ cos.a; — cos.(y—x). (7) 
And as before suppose y=:nx, (7) will become 

cos.(7i-|-l )x=^2(ios.nx cos.a; — cos.(w — 1 )x. 
Making 7i=l, 2, 3, 4, (fee. in succession, we shall find 
cos.2a;=2oos.*a? — 1. (8) 



1 



MULTIPLE ARCS. 307 

cos.3a?=4cos.3a; — Scos.a:. (9) 

cos.4a;=8cos.''a; — Scos.^ar+l. (10) 

cos.5.r= 1 6cos. ^o: — 20cos. 3a:-|-5cos.«. (11) 
&c. &c. 

By the aid of the well known equation, sin.^a;-|-cos.^.'ri=l, com- 
bined with equations (3), (4), <fec. to (11), as circumstances 
may require, we obtain 

sin.iP=sin.a:. (12) 

(8) substituted, sin.^a;=l — cos.2ic=|(l — cos. 2a;.) (13) 

(4) transposed, sin.3a;=i( — sin.3.r-j-3sin..'r.) (14) 

(13)2, (^0) & (8) sub. sin.^a;=|(cos.4.r— 4cos.2a;+3.) (15) 
(6) reduced by (4), sin.5a;=yV(sin.5:i; — 5sin.3;?;-|-10sin.a;.) (16) 

If we multiply equations (12), (13), (14), (16), and (16), 
by dx, and integrate, we shall have the following series of equa- 
tions : 

Csin.x dx= — cos..r. ( 1'^) 

fsm.^xdx= — \sm.2x-\-lx. (18) 

Jsin.^xdx=\.^^-—lcoa.x. (19) 

fs'm.''xdx=},^^-—{sm.2x-\-^x. (20) 

*- 4 

/r J , cos.SoJ , 5 cos. 3a; , „ /m \ 

sm.'xdx=—f\ _|-_5-. __i.|cos.a;. (21) 

o o 

(fee. (fee. 

(Art. 92.) If we turn back to equations (8), (9), (10), (fee. 
in the last article, we can find a series of equations expressing 
the powers of the cosine of x, as follows : 

cos.a:=cos.a;. (22) 

2cos.2.r=cos.2ar+l. (23) 

4cos.3a;=cos.3a;-]-3cos.a;. (24) 

8cos.^a:=cos.4a;-|-4cos.2a;-|-3. (25) 

(fee. (fee. 



308 INTEGRAL CALCULUS. 

Multiplying by dx and integrating, we have 

J*co8.xdx=8m.x. (26) 

rcos.^xdx=\sm.2x-{-^x. (27) 

rcos.3a;c?a;=yVsin.3a?-f-4sin.a;. (28) 

rcos.^xdx=~^2^mAx-]-lsm.2x-\'^x. (29) 

(Art. 93.) We may also integrate circular functions in the 
tbllowing manner. We give but one example, which is intended 
as a general illustration. 

Integrate cos.^xdx. Place cos.a;=^. 

Then <?;?=~-J^==--_-^. But cos.^aj^^s. 



sm.a: 



Jl—z' 



Whence Ccos.^xdx^ — f _, and the integral of this 

last expression is to be found in the second and third equations 
of the first example in (Art. 79), which is 



Replacing cos.a; for 2;, and we have 

fcos.^x dx=^eos.^xJl — cos.^ic+f ^1 — cos.^ar. 

The integral of this same quantity is found in (28) of the 
last article. Therefore 



icos.^o?^! — Gos.'^x-^-jJl — cos.2.'c=yVsin.3^4"|si^'^' 
but this result reveals nothing new. 

In a similar manner we might integrate many of the differen- 
tials in the preceding article, and thus find many other equations 
between sines and cosines. 

(Art. 94.) We may integrate differentials in the form 

sin.ajcos.^iccfe by the same general principles. 

Assume cos.a:=^. Then, as in (Art. 93), 

„ , z^dz 

QOB,^xdx=. — ... 



SUCCESSIVE INTEGRATIONS. 309 



But sin.a;=^l — z- ; therefore Bm.xQos.^xd.i:= — z^dz. 
And jB\n.x.coB.^xdx=^ — — -]-(7. 

In general terms rsin.a;cos.°c^a;= — = — — - — -\-C. 

The same general principle will enable us to integrate a dif- 
ferential in the form 

dx 
sin.3.r 

Place 8in.a:=^. Then dx= — —, and 



cos-.r sm.^a; cos.ir.2^ 



Whence r_^_=: r_ ^^ 



sm.**a; ^ ^ J\ z^ 

This last differential has already been integrated by (Formula 
c?), example 3, (Art. 79.) 



CHAPTER IX. 
Successive Integrations. 



(Art. 95.) In the diflferential calculus we perceive that every 
equation of the first degree between two variables is susceptible 
of being differentiated but once. An equation of the second 
degree can be differentiated twice. An equation of the third 
degree three times, and so on. 

For instance, if y=sa.r ^ -f-5ar^ -\-cx-\-d. ( 1 ) 

We shall have by successive differention 



dx 


(2) 


Jf=e«.+.. 


(3) 


s=- 


(4) 



310 INTEGRAL CALCULUS. 

Another differentiation and division by dx, would give 

dx'' 
Hence, if n be the degree of an equation, the (1+%) differ- 
ential coeJ3icient will be zero, unless the independent variable is 
a reciprocal quantity, as in the following example : 

X dx x^ dx^ x^ dx^ x'^ 

Now by inspecting the preceding examples it is obvious that 
the wth differential of y divided by the ?^th power of the differ- 
ential dx, must be equal to X, C, or 0, that is, some function of x, 
represented by X, or to a constant quantity C, or to 0. 

Successive integration is the converse of successive differenticds, 
and to illustrate the operation, we will take equation (4) in the 
first example, and integrate it. 

dx^ 
Multiplying each side by dx, and we shall have 

^ly=zQadx. 

Now, regarding all powers of dx greater than the first, as 
constant, and integrating, the first integral will be 

dx'' ^ 

Multiplying again by dx, and integrating as before, the second 
integral will be 

"^^^ax^-cx+C. 
dx 

And again, and the final integral will be 
y=ax^-\.—x'' +c'x+ G". 

This is the same in form as equation (1), and would be iden- 
tical if we could restore the identical constants b, c, and d in 
place of c, c, and C". 

In ato'ftc^ examples, particular constants cannot be determined. 



SUCCESSIVE INTEGRATIONS. 31! 

If we have ^=0, the first integral will be 

That is, some constant; and the final integral will be 
y=lCx'J^C'x+C". 

If we have zJiz=X, JT being any function of x, the first in- 
dx^ 

tegral will be 

fj.==JXdx+C, 

UiX 

The second, ^=JdxJXdx+Cx+C\ 

The last, y=JdxJdxJXdx+^Cx^'\^€'x+C". (A) 

This last equation may be indicated thus : 

y=J^Xdx^+\Cx^'^c'x+c\ (B) 

The sign C^ indicates three successive integrals. 

(Art. 96.) A differential in the form 

d'^v 

— |.=a+y2, or =2/2^ or =3^^, 

Or in general equal Y, J^ being any function of y, can be inte- 
grated as follows. Multiply each member by 2dy. 

Then '^y-'^'l=.Ydy. 

dx . dx 

The first member, we perceive, is the differential of _^, on 

dx^ 

the supposition that dx is constant and dy variable, (and dy is 

always variable, or we could have no second differential oi y), 

therefore, by integration, we have 

Or ^- =(fo. 



312 INTEGRAL CALCULUS. 
Whence x= f -"^J^ + C'. 

For a particular example, we require the integral of 

d^y y Idy d^y ^ydy 

^2~ ^' '~dx'~dx a^ 



dx^ a^^ ' dx V a^' 

dy 



Jc-yr • 

f — " r- ady /• dy 

' ft 
If we make Ca^ = l, then C7=— 2, and the last integral be- 



comes 

asin.~*y-|-C". Or a;=a. sin.~»y-(-(7' 
For a second example, we give the following : 
d^y __Py 
dx"" b ' 



This is solved on pages 350, 351, of the author's Operations. 



3. Integrate the differential -i 'T " / =:a. 

dxd^y 

In the differential calculus we put -^=». 

^ dx ^ 

Whence ^^=^. 

dx^ dx 

These values substituted in the given differential, will reduce 
It to 

dp 



GEOMETRICAL INTEGRALS. 313 

Whence dx= — —^ — , and pdx=dj/= — ^^ — 

Integrating the last two expressions, we obtain 

and eliminating jo, we find 

(ar-C)^+(y-.(7')2=a^ 

the general equation of the circle, C and C being the co-ordi- 
nates of the center. 

This result was expected, because the given diflferential cor- 
responded to a constant radius of curvature. See (Art. 51.) 



CHAPTER X. 
Oeometrieal Integrals. 



(Art. 97.) In this chapter we propose to show the applica- 
tion of the integral calculus to geometry ; an operation of the 
greatest utility in finding the measure of surfaces and solids. 

In (Articles 53, 54, and 55), we have shown geometrical 
differentials, and by the integration of these, we shall have the 
corresponding integrals, which will be the measure of surfaces 
or solids, as the case may be. 

To commence with the most simple 
case, we require the area of the space 
NPM, one side of which is hounded hy 
the curve NM, o% the supposition that 
the curve is a portion of a paraholay 
and the point N its vertex. 

Let iV^be the zero point, N'P^=x, 
PM—y, and it is obvious that ydx 
is the differential of the area required. 

By the equation of the curve we have y3=2/>«. 




314 INTEGRAL CALCULUS. 

Whence ydy=pdx, ydx=:^^ — ^. 

When a;=0, y=0, and therefore (7=0, and the whole integral 

That is, the area of any portion of the parabola bounded by 
the axis and ordinate is measured ly two-thirds of the rectangle 
made by the abscissa and ordinate. 

The same was shown in analytical geometry. 

(Art. 58.) M>w let the area of the same space he required on 
the supposition that the- curve is a circle and the radius unity. 

The same notation as before. The equation of the curve is 
(l-o;)^ +2/^ = 1. (1) 

Whence Jydx==:jt^=.J 



The part — ===: expanded into a series by the binomial, 
and afterwards multiplied by y^, produces 

y^+r+^l+^yl_+hMyll.-\- &c. 

^2 2.4^2.4.6 ^ 2.4.6.8 ' 
Multiply each term by dy, and integration will give 

rsrs Sy^ . 3.5y^ , 3.5.7y»^ ^^^ .^. 
3 ' 2.5 ' 2.4.7 ' 2.4.6.9 2.4.6.8.11 
This integral requires no correction, or rather, the value of 
the correction is 0, because x and y vanish together. 

Here ^ is the sine of the arc NM, and it may be of any value 
from to 1 . 

When y— 1 , the space NPM will represent a quadrant, and 
its numerical value will be the sum of the series 

3 ' 2.5 2.4.7 ' 2.4.6.9 2.4.6.8.11 
Four times this series is the value of the whole circle. 



GEOMETRICAL INTEGRALS. 315 

When we make y=l, series (2) does not converge with suffi- 
cient rapidity to meet with practical favor, therefore we assume 
y=i, knowing that 3/=^ when the arc is 30°. 

Substituting y=^ in series (2), we have for the area of the 
half segment NFMoi 30°, the following series 

3.23^ 2.5.25 « 2.4.7.2'^ 2.4.6.9. 2» 2.4.6.8. 11 .2^ » 

The sum of this series carried sufficiently far, will be found 
to be .04529302, the double of this, or .09058604, is the area 
of a segment containing 60° of arc in a circle whose radius is 
unity. 

In any other circle whose radius is r, the area of the segment 
containing the same number of degrees is (.090586 ?'2.) 



(Art. 99.) To find the area of the whole circle by the aid 
of this semi-segment .04529302, we will turn back to the figure 
and conceive a line drawn from M to the center of the circle. 
The hypotenuse of the triangle so formed is 1, and the perpen- 
dicular y=^, therefore the base is ^3, and the area of the tri- 
angle is .21650637. 

Whence to NPM 04529302 

Add the triangle \ . 1^3 21650637 

Sum is sector of 30° 26179939 

Multiply by 12, 12 

Area of the whole circle, 3.14159268=^ 

To find the circumference of this circle we will for the mo- 
ment represent it by x, and the radius by r. But the area of 
any circle is the product of the circumference into half the 

radius, therefore -1-=^, and x==2n on the supposion that r=\. 

Hence, when the radius is I, the length or measure of 180° 
of the circumference is 3.14159268, the most important number 
in mathematical science. 

(Art. 100.) Take the same figure as before, and conceive 




316 INTEGRAL CALCULUS. 

NMio be a portion of an ellipse, N" being at one extremity of 

the greater axis. 

As before, let N be the origin of 
co-ordinates, J}fP=x, PM=i/f and 
from iTto the center of the circle, or 
to the center of the ellipse, we will 
designate by A. Then, if the curve 
is a circle, y=J^Ax — x^ . If the 

curve is an ellipse, ^=-7- ^/ ^Ax — x^> 

£ being the semi-minor axis of the ellipse, according to custo- 
mary notation. 

Now we have just seen that the area of a circular segment is 
represented by 

Cydx=Cdxj2Ax — x^. {a) 

The area of an elliptic segment on the same abscissa x, and 
corresponding on the same major axis ^A, must therefore be 
represented by 

^JdxJkAx—x^ . (b) 

These segments are any like portion of the circle, and the 
ellipse; when a;=^ the segments will be quadrants. But like 
portions of any two magnitudes are to each other as the wholes 
are. Therefore 



TO 

area circle : area ellipse : : fdxj2Ax — x^ : — CdxJStAx — x^ 

72 

Or area circle : area ellipse : : 1 : . 

But the area of the circle in question is =/t^2. 
Whence nA^ : area of ellipse : : 1 : — 

Or Area of ellipse=;<^5. 

But TtAB is obviously the area of a circle whose radius is 
J AB, a mean proportional between A and B. 

N. B. This article is the same as propositions 10, 11, and 12, 
of the ellipse in Robinson's Geometry. 



> 



GEOMETRICAL INTEGRALS. 317 

(Art. lOL) In this article we propose to find tlie area eii- 
closed by the cycloid. 

Let r be the radius of the generating circle, NA the axis ot" 
X, and xVthe zero point. 

According to custom, place iVP=a;, and PM^=y, The object 
is to find the double area NAB. 

The differential of the segment 
NPM is ydx, hence the integral of 
this, is the segment itself: and if we 
suppose X to be equal to JVA, the 
segment will be the area of JVAB. 

The differential equation of the 
cycloid is 

dx= y^y (Art. 48.) (1) 

Whence Jydx=J—£M=:=.-=jSfPM-\-C. 

J^ry—y"" 

This differential has already been integrated in (Art. 80), it 
is the fourth example. All we have to do is to change x to y, 
and a to r, in that example, and the result is 

_A_|_3^\ ^2ry— 2/2"+^ vers. sin.-^^+C. 

But when we make NPM^^O, y=0, and therefore C=0. 
When we take y=9.r, the above result becomes 

37*2 . _, 2r 3?rr2 

— vers. sm. ^ — = 

2 r 2 

The double of this, (S^r^), is three times the <jrea of the gene- 
rating circle, the area required. 

We can, however, obtain this result by a more simple process, 
if we take into consideration the external space DBK. 

The rectangle ABDN" \s obviously (NA).(AB), which is 
(rtr)(2r), or 27tr^, and now if we can obtain the value of JSfDB, 
and subtract it, the remainder will be the area NAB. 

The differential of the space NDEM is {EM)dx. But 

EM=2r—y. 

2.J. 



318 INTEGRAL CALCULUS. 

Hence d.NDEM= ( %r—'y)dx. 

If in this we substitute the value of dx from (1), we shall 
have 

d.NDEM= i ^''—y)y^ ^dyJ^;^-2 . 

Whence, by integration, N'DEM=Cdyj2ry—y^ . 

By comparing this with expression (a) in (Art. 100), we per- 
ceive that the second member is the area of the segment of a circle 
whose radius is-r and abscissa y. 

That is, NMEB^ALH. 

Hence NBD=^ the semi-circle AHB. 

But the area of the semi-circle is .. 

2 

Ttr^ 

Whence AN'B=^Tcr^ — , and the double of this is S^r, 

2 

the area sought. 

That is, the area of the cydoidical space is three times the area of 

the generating ciyxle. 

(Art. 102.) While on the cycloid, let us require the value 
of any portion of its arc, as BM. 

Let s be the length of an arc, whatever be the curve; then we 
have seen in (Art. 65), that 

ds=Jdx^+dyK (1) 

In short, this equation is obvious in a primary point of view. 

But dx= ^ ^ , and this value put in (1), produces 

sj2ry—y^ 



ds=J_yl^_+dy^ = dy l_^_, . . 

N 2ry—y^ ^ 2r — y ^ ' 



By integration, s=—2j2r,j2r—y-\-0. (3) 

If we estimate the arc from B, making z=0, then y=2r, and 
0=0-f-(7, which shows that there is no constant to add, and 

s=—2j^,j2r-^y, (4) 

which is the general expression of any arc estimated from B, 



1 



GEOMETRICAL INTEGRALS. 319 

If BM=s, the corresponding value of y is AL, and 2r — y= 
BL. Hence, 



BM=—2JAB.JBL=—2JAB,£L. 



But JAB.BL=dzBir. Therefore BM=2BIf. 

That is, The arc of a cycloid estimated from the vertex is twice 
the corresponding chord of the generating circle : And the arc BMN 
is tioice the diameter of the generating circle, and the entire cycloidical 
arc is four times the diameter of the same circle. 

It is necessary to have a formula for the circumference of the 
ellipse, — we will therefore substitute the values of dx and dy, 
drawn from the equation of the ellipse in the general formula for 
the arc of a curve, that is, in Jdx^-\-dy'^, and integrate. 

Let the center be the origin of co-ordinates, and the equation 
of the ellipse is 

For convenience Ave shall use the eccentricity of the ellipse, 
which is the distance from either focus to the center, when A=l, 

A2 J52 »2 

or c^= , whatever be the value of ^. Then 1 — e^= 

A^ A^' 

Whence y^=(l—e^ )(A^—x^ ). 



dx 



y jA^—x 



In short, ds=^ J dx^ -\-dy 



_ AdxJ 



1_^^ 



JA^—x' 



Expanding l\ ^^^ into a series, we obtain 

ds=—^^^^^t\—^^^^—^^^^ — ^^^^ — (fee ^ 
^~ JA^—x^\ 2^^ 2AA^ 2.4.6^6 / 

Multiplying each term of the series by the factor without the 
parenthesis, we shall have a series of differentials which can be 
integrated separately, one by one, which together, will show the 
approximate value of any arc corresponding to any assumed value 



3|2G mTEGRAL CALCULUS. 

of X less than ^. When we make x=A, after integration we 
have one-fourth of the circumference, which multiplied by 4, 
gives the whole circumference, which is 

%nA(l-^ ?fL_ ^^^ &C. ^ 

\ ^..2' 2;2'.4.4 2.2-.4.4*6.6 / 

(Art. 103.) The curvature of the circle is uniform, and as 
we have found the value of the whole circumference (2^), to 
radius unity, therefore we shall have, by simple division, the 
value of any required portion of it. But it is not so with other 
curves. 

To find the lengths of other curves between any proposed 
limits, we must integrate the expression Jdx^'\-dy^ , taking x or 
y between the proposed limits, (the equation of the curve given,) 
and the relation between x and y in each particular case. It is 
not necessary that we should give examples in every curve, we 
will therefore select the most interesting, as one of the spirals. 

The general equation for the spirals is r=a^", (Art. 43,) in 
which r is the varying radius, t the measuring arc, and (X a con- 
stant quantity. 

The differential equation of an arc in respect to polar co- 
ordinates is 

d»z=zjr^dt^-fdr^, (Art. 41.) 

therefore the integral of this expression is s, or the length of a 
spiral curve between proposed limits. 

From the equation of the curve, dr'^^^^n^a^t^^'^dt^ . 

Whence Jjr^dt^-\^r^ =Jdi(t^+n^)hir-K 

When %=1, as is the case in the spiral of Archimedes, the 

diflferential becomes 

aju\^.dt\ 
By the ninth example (Art. 80,) we find that the integral of 

this is 



</14:^+^ log.(^+Vl+^'')+^- 

N. B. We shall find an expression of the same form for ao 
arc of the common joara6o/a. 



GEOMETRICAL INTEGRALS, 321 

If we take the logarithmic spiral whose equation is t=\og.r, 
we shall find 

Whence rJ^-\-C, (or simply rjlt, commencing at the origin 
of the radius vectors,) expresses the arc of this curve, **and Ave 
see that though there is between this origin and any point of the 
curve at an infinite distance from it, an infinite number of revo- 
lutions, yet they include an arc of finite length, which is equal 
to the diagonal of the square described on the radius vector." 

AREA OF SPIRALS. 

(Art. 104.) In (Art. 41) will be found the differential of a 

polar sector, or the difterential area of a polar curve, equal to 

in which r= when we apply it to the spiral of Archimedes. 

Whence f- — = fr(r^dr= = . 

If we assume (=^27t one revolution, 
this last expression for the area corres- 
ponding will be -, showing that the 

space PMA included in the first revo- 
lution from the pole is equal to one- 
third the area of the circle, whose 
i-adius is equal to the radius vector at the end of the first 
Involution . 

If we make if=27t, the area described in two revolutions is 

* ^ , but this includes the first revolution described the second time^ 
3 

hence the area actually enclosed after two revolutions, will be 




322 INTEGRAL CALCULUS. 

Again, if we take the logarithmic spiral whose equation is 
t=\og.r and apply it in (1), we shall have 

^r^dC ^rdr '^^ _\ri 

J ~2 *^ ~2 4 "" 
If we estimate the area from the pole where r=0, which makes 
(7=0, and the whole area is — , that is, the area of the Naperian 

logarithmic spiral is equal to one-fourth of the square described on 
the radius vector. 

Again, if we take the hyperbolic spiral, then r= — 1, and the 

general equation of the spirals becomes r= — 

t 

Whence fJ^=- [^1=-"^. But -^=-^'. 

This area is infinite when ^=0, but we can find the area in- 
cluded between any two radius vectors, b and c, by integrating 
between the limits t=^b and /=c, which will give 



9,\b c) 



CHAPTER XI. 
Oeometrical Integ^rals, continued. 

THE AREAS OF CURVED SURFACES, OR SURFACES OF REVOLUTION, 
DETERMINED BY INTEGRATION. 

(Art. 106.) If any curve, as NM, 
revolve on an axis, as NP, the axis 
of X, it will describe a curved surface. 
If the curve is a circle, the surface 
so described will be the surface of a 
sphere. If the curve NM be a por- 
tion of a parabola, the surface it will 
describe will be a parabolic surface, &c. &c. 




GEOMETRICAL INTEGRALS. 32^ 

If N^Afhe a straight line, the surface described by revolving 
on the axis of Xwill be the surface of a cone, or if a portion of 
the line only revolve, the surface so described will be the surface 
of a conic frustrum. 

It is obvious that we must obtain the general differential ex- 
pression for these surfaces, and it is obvious that this diflference 
is measured by the revolution of a small portion of the arc at 
M. Or, by the revolution of the differential of the arc, which is 



And this line revolves at the extremity of the radius y. 



Therefore ^ny J dx^ -\-dy^ is the differential in question. In 
words. The differential of a surface of revolution is equal to the 
circumfereTice of a circle "perpendicular to the axis, into the differen- 
tial of the arc of the meridian curve. 

Our first application of this formula will be to the circle, be- 
cause most persons are more familiar with that curve than with 
any other ; — therefore 

1 . Required the surface of a segment of a sphere corresponding 
to the co-ordinates x, y, the origin being at the circumference. 
Let i2 be the radius : Then {R—xY-\-y'^=R''. (1) 

From which we find Jdx^+dv^= ^^^— . 

JR'-y' 

Whence ^7iy Jd^^d:^ = lll^^^. 

JR-^-y^ 

C%7tyJdx^+dy-z=2,iR. f^ l^ =—'^7t R JR^ — vM- C 

JR'-y' 

We perceive by the figure, that commencing at N, the zero 
point, making a;=0, 2/=0, and the area in question equals 0. 
That is, the equation above becomes 

0=—27iRjW-^C. 
Or C=2jiR^. 

Substituting this value of C and the general integral becomes 



—27iRJR^—y^-^-27iR' 



324 INTEGRAL CALCULUS. 

When we take y=M, the surface corresponds with a hemis- 
phere, and the factor JH^ — y'^, then becomes 0, showing that 
the surface of a hemisphere whose radius is B, is 2ftE^ . There- 
fore the surface of the whole sphere is 47tB^ . 

Again, when we take x==2B, y will again be 0, and the first 
term of the expression for the general integral becomes 

showing that the area of the surface of revolution is 

±2rti22+2rt^2^0, or 4;ti22, 
corresponding to y=0, the latter value is the surface of the whole 
sphere. 

(Art. 106.) Conceive NM to be a straight line, then the 
area of the surface of revolution will be the surface of a cone. 

And in that cone we shall have 

X : y=a : b. Or x=~. 

^ b 



Jdx-+dy^=^^a^.^a^.^^^a^+b'^ 

.2^ rw.,^ njj-n^-^^y' - 



f2xyJdx'-\-dy'=±^Jydy{Ja^+b'^)=^(a'+b'y- + C. 

If we conceive the area to commence at the point JV where 
(=0, we shall hare the area equal 0, and y=0, which will give 
(7=0, and the whole integral will be 

b 2 



But Ja^~{-b^=^M', and if we make y=b, the surface will be 

an expression which is obviously ike circumference of the base of 
a cone midtiplied by the half of its slant hight. The same rule as 
was found in geometry. 

When the curve NM\^ a parabola, the surface of revolution 
is called 2i paraboloid. 

In that case we have y^=^2px, ydy=pdx* 



GEOMETRICAL INTEGRALS. 325 

Whence Vrf^M^^-^S^^yI='^ VF+^- 



2^ r..^.. /:.^~r;,"2_2rt/.,2 



When y=0, the surface of the revolution is 0. Therefore 

Or (7=_2-^. 

3 

Whence the entire integral between the limits y=0 and 3/=^, 
may be written thus : 

|((*=-k'=)-.=)). 

When the curve is an ellipse, the result comes out in a series 
more tedious than interesting. 

When the curve NM is a portion of the cycloid, we have 



V2ry- 



Whence dx^J\-dy^- ^yl^yl_ J^dif = ^Z^^!. 

2ry — y- 2r — y 

And '^^yJd^-\^dy'' = '^7ij¥r(—^^\ ( 1 ) 



Whence J 'ZTty J dx"" -\-dy^ :='^7i J 2r J . 



ydy 



To integrate this last expression, place ^2r — y=^z. (2) 
Then y^=^r — 2", and ydy=^ — '^rzdz-^^z^dz. 

Substituting this value in (1), restoring the value of (s) at 
the same time, and we shall have 

2.^72^/— 4/V2>-— 3/+f (2r--y)'^^+6'. 
To find the value of we must consider that wlien ?/=0, the 



326 INTEGRAL CALCULUS. 

surface sought must be 0. Hence, the following equation must 
be true : 

0=2^(-Br^+^)+(7. 

Whence C= . 

3 

If we make y=2r, the value of half the surface sought is nu- 
merically the same as C, because ^2r can be taken with the 
minus sign. Hence, the whole surface equals %^7ir^, which is 
sixty-four thirds the area of the generating circle. 

The preceding examples are sufficient to illustrate the theory 
of finding the area of surfaces by integration. 



CHAPTER XII. 
Oeometrical Integrals, continued. 

THE VOLUME, OR CUBATURE OF SOLIDS OF REVOLUTION. 

(Art. 107.) The motion of a line is conceived to form a sur- 
face, and the motion of a plane, or the revolution of a plane on 
an axis, may be conceived to form a solid or a geometrical 
volume. 

But it is not necessary to conceive a revolution of a plane 
to obtain a solid ; we can take a solid like a parallelopipedon, 
a cone, or a pyramid, and conceive it to inc7'ease or deci^ease 
by the motion of one of its surfaces, and thus we have its dif- 
ferential. 

The integral of this differential, corrected if necessary, will 
give the volume to that differential. We shall give both 




GEOMETRICAL INTEGRALS. 327 

methods, calling particular attention to the first problem, on 
account of its simplicity and its elementary character. It is this : 
1st. Find the volume of a pyramid, by integration. 

Let G be the vertex of a pyramid, and 
assume GF=1, and designate the corres- 
ponding base FOHE by h. Let 6^-4=0?, 
and AIQM, the corresponding base. 

But these bases are in proportion to the 
squares of the distances from the vertex. 

Therefore 1 : a;^ : : b : bx^=AIQM. 

The differential of this pyramid is obviously ha^ dx. 

Hence the pyramid itself is Cbx^dx. 

Bu Jbx'dxz=^+C. 

This is true for all values of x, it is true then when x=0, and 

making this supposition, the last equation becomes 0=0-\-C, or 

bx^ 
(7=0. Therefore is the whole integral, or the solidity of 

the pyramid. 

But =:(bx^)-. That is, the base multiplied by one-third 

of the altitude gives the cubical contents of a pyramid. 

N. B. When the base is a circle the pyramid becomes a 
cone, to which the same rule applies, namely, 

T/ie area of the base multiplied by one-third of the altitude. 

Scholium. A sphere may be conceived to be composed of a 
great multitude of pyramids, the base of each one being a very 
small portion of the surface of the sphere, and the altitude of 
each one the radius of the sphere. Therefore, the volume of a 
sphere is equal to its surface multiplied into one-third of its radius. 

Again, we may conceive the triangle GAI to revolve on the 
axis GA, thus forming a cone. The radius of the base of that 
cone will be AI, which we will designate by y, then Tiy^ will be 
the area of the base, and ny^dx will be the differential of the cone. 

Hence the cone itself will be Cny^dx. 



328 



INTEGRAL CALCULUS. 



We can integrate this, provided we can find the relation be- 
tween X and y. 

Let GF=1, and FE=a, GA being x. 

Then by proportional triangles, 1 : a=x : y. y=zax. 



,x=--x. y^ 
3 *^ 




Which is fny- .- Y the area of the base multiplied by one-third 
of the altitude. 

(Art. 108.) Required the volume of any solid of revolution t 
as the segment of a circle, a segment of a paraboloid y the segment of 
an ellipsoid, <&c. 

Let iV be the origin of co-ordinates, 
H the mdius of a circle, NP=x, 
PM=y as before. 

Now it is obvious that the revolu- 
tion of the segment JSfPM, on the 
axis NPy will produce a solid, and it 
is also obvious that the revolution of 

PM, (ydx), on the center P, is ny^dx the differential of the volume. 

Hence, the volume of revolution between the limits a;=a and 

.r=^, is found by 

For the segment of a sphere we have the equation 

From which y^ = 2Rx — x^ . 

Whence 

jTty'-dx=J{%7iRxdx—7ix^dx)^7tRx^—-l-{. C. 

When .^=0, the area is 0, therefore C=0, and the integral 

TtRx^ — V the solidity 0/ 

the segment. 



GEOMETRICAL INTEGRALS. 32^ 

When x=jR, the segment is a hemisphere, and its solidity is 
'iitH^, and when x=2li, or when the segment contains the whole 
sphere, its volume is ^TtM^ , the same result as was found in ele- 
mentary geometry. 

This result also corresponds to the scholium in (Art. 107), 
for in (Art. 105) we found the surface of a sphere to be 4rtE^ , 

which multiplied by — produces , the solidity of the 

sphere as before. 

(Art. 109.) We may change the origin of co-ordinates from 
the surface to the center of volume at pleasure, or we may in 
fact change it to any other known point. 

For example, we will recompute the last problem, taking the 
center of the sphere for the zero point, 

III that case x''+y''=B\ or y^^E'^—x'' . 

Whence f Tty^ dx= f (rtRdx — Ttx^ dx)=^HR^ x — -f"^- 

When ir=0, the volume =0, and therefore 6'=0. 

When x=Ri the volume corresponds to a hemisphere and the 

expression to hR^ — = , the same as before. 

3 3 



JtX 



N. B. In the expression nR^x — , x cannot be taken 

greater than R. In case it be so taken, the numerical value of 
the whole would be minuSy but magnitudes cannot be essentially 



Lei us now require the volume of an ellipsoid, the ellipse revolving 
on Us major axis. 

The equation for the ellipse is 

A^y^+B^x^=:^A^B\ 

■ Whence y^-=^B^ — "la"*"' 






j7ty^dx=J{rtB^dx—H^^'^dx)= rtB^x—n^^- ^' 



330 INTEGRAL CALCULUS. 

The origin being the center when x=0, the volume corres- 
ponding equals 0, and consequently (7=0. 

Hence, the A^alue of any segment of an ellipsoid must be 



TtB^xf 1 



\ 3^V 



If we make x=A, the expression will correspond to a semi- 
ellipsoid, and it will reduce to 

frt^M, or piB'-4A. 

That is, two-thirds of the circumscribing cylinder. 

If we suppose ^=-S, the ellipse will become a circle, and the 
semi-ellipsoid will become an hemisphere, and the expression 
above will become f 7ti2^, as it ought to do. 

(Art. 110.) If an ellipse revolve on its minor axis, it will 
describe an ohlate spheroid, and the differential of the volume 
will be 

Ttx^dy, 

Butx^ = A^'-'~-y\ Whence rtx''dy=jt^A''—^y''\dy. 

J,,^dy=.A^y-!Lf^, 
If we make y=B, this solid will be expressed by 
tytA^B—'^^\ or iB,7tA\ 

This is also two-thirds of the circumscribing cylinder. 
Comparing the two solids generated by the revolution on each 
axis, we find 

oblate solid : prolate solid : : BA^ : AB^ 

:: A : B. 

To find the volume of a paraboloid, we have the equation of 
the parabola y^ =9.px. 

This value of y^ placed in the general expression J ny^dx 
will give us 9,jtpxdx for the differential of this solid. Hence the 
solid itself is :tpx^ , which requires no correction. 



GEOMETRICAL INTEGRALS. 331 

But rt.px^=. — ?-- .x^^ny^ . -, and this we perceive is one-half 
of the circumscribing cylinder. 

Scholium. Let y be the radius of a circle, then its area will 
be Tty^ . Let h be the altitude of a cylinder. 

Now conceive a cylinder, a cone, a paraboloid, and a sphere, 
to equal circumferences, and each the same altitude h. By a 
short retrospect we find 



The volume of a cylinder 


=rt3>= .h. 


The volume of a cone 


2 A 


The volume of a paraboloid 


=..|. 


The volume of a sphere 


=..L^. 



Calling the cylinder 1, we have for the cone ^, for the parabo- 
loid \, and for the sphere |. 

The proportion in whole numbers is, cylinder 6, cone 2, para- 
boloid 3, sphere 4. 

These proportions were discovered by Archimedes, and it is 
said that he requested them to be engraved on his tomb. 

For another example, we require the volume generated by the revo- 
lution of a cycloid on its base. 

The differential of any revoloid is jty^dx. 

For the cycloid we have dx= — J^ ^ (Art. 48.) 

J2ry—y'^ 

Whence J ^y- dx= J -^£M=.^=^ J -^^M==-. 

J^ry—y^ J^ry—y'' 

This is integrated by formula d, (Art. 78,) as follows : 
Jlry—y'' 2 2 Ji^ry—y' 



332 



INTEGRAL CALCULUS. 



/- 



2/% _ 



J9.ry—y'' 



J2ry—y^ 



J'^ry 
=arc( 



'+'-/-T^: 



dy 



(3) 






ver. sm. 



J^ry^y^ 
\ (4) (Art. 79.) 




These integrals require no con- 
stant, for when we make 2/=0 the 
volume will be 0, as it ought to be. 

If we make y=2r, the corres- 
ponding volume will be half the 
volume sought, and (4) will be- 
come rt. 



fhis value put in (3) will give 



■Tir. 



And this substituted in (2) will give 

r y^d y ^ ^nr^ 

J^iry—y'^ 2 
And this placed ii^ (1) and multiplied by n produces 
r y^dy _5rt^r^ 
J2ry—y^ 2 

This being half the volume sought, the whole must be 

But n(2ry represents the base of the circumscribing cylinder. 
And 2rtr represents its altitude. , 

Therefore STt^j-^ is its solidity. 

Hence, the solid required is Jive-eighths of (he circumscribing 
cylinder. 

(Art. 110.) Now conceive the curve to revolve on the axis 
of Y, iVthe center of revolution, and iVP the radius. On the 
supposition that A^PJfis a portion of a parabola, we require the 
volume generated by the revolution of the curve on the axis of 
T", the origin being at N'^ or the axis being changed from A 
ta K 




GEOMETRICAL INTEGRALS. 333 

To solve this problem we must ob- e^b^^^^^^^^^bhb < 

tain a new and corresponding expres- ra^^^^^^^^^^^BB i 

sion for the differential of the volume. 

It is obvious that ydx is the differ- 
ential of the revolving surface. Thi^ 
revolving at the extremity of the ra- 
dius X, will revolve through a space 
equal to ^nx. Hence, the differential 
of the volume of revolution will be expressed by 

9,7ixydx. 

This may be applied to any curve (revolving on the axis of Y 
and center iV", ) as well as to the parabola. We take the parabola 
because the integral comes out in a definite form. 

The equation of the parabola is y^=:2px. 

Whence x==^. And ^7txydx=!!^^^ 

But this volume requires no correction, for when ar=0, y=0, 
therefore (7=0, and the volume sought is 

!*^. But y*=4p^x'. 

Whence !^=i!^.^= F, the volume sought. 

Let us observe that rtx^ is the base of the circumscribing cyl- 
inder, and y being its altitude, nx^y is the volume of the cylinder. 
Now by proportion, 

cylinder : F ; : Jtx^y ; -• 

5 

: : I : 1 
5 

Whence F=f of its drcnmscrihing cylinder. 

Scholium. Hence the volume around the axis of Abounded 
by a- portion; of a parabola, is one-fifth of its circumscribing 
cylinder. 

22. 



334 ■ INTEGRAL CALCULUS. 



CHAPTER XIII. 



On the Integration of Homogeneous and l,inear 
Differentials. 

(Art. 111.) An equation is said to be homogeneous when 
the sum of the exponents of the variables is the same in every 
term. 

Differentials of this form can always be integrated. In such 
cases we place one of the variables equal to the other multiplied 
by an assumed variable factor, but we shall illustrate by 

EXAMPLES. 

1. Integrate the differential 

x^ dy=y^ dx-\-xydx. 

This equation is homogeneous. Therefore place x=zvy. Then 
the equation becomes 

v^y^dy=iy^ dx-\-vy^ dx, 
which is divisible by y^ , and 

v^ dy=dx-\-vdx= ( 1 -|-v )dx. ( 1 

But x=vy. Therefore dx=vdy-\-ydv. (2 

The value of (dx) substituted in (1), it becomes 
V 2 dy=:vdy-\-ydv-\-v^ dy-\-vydv. 
Or 0=zvdy-\-ydv-\-vydv. (3) 

Dividing each term by (vy), we have 
dy , dv , J ,. 
y V 
By integrating each term, we obtain 

log.y-j-log.'y-(-v= 0. 
Or \og.(vy)-\-v=C. 

That is, loff.a;4-^=C, the result sought. 



PARTICULAR INTEGRALS. 336 

2. Integrate the differential 

{x^ -\-xy)dy=(x—y)ydx. 
As before, let x=vy, then 

( v^ y^ +vy^ )dy—{vy—y)ydx. 
And {v^-\-v)dy={v^\)dx. (1) 

Because x=vy, dx=zvdy'\-ydv, and this value of dx substituted 
in (1), produces 

(v^^v)dy=^v^dy-\-vydv — vdy — ydv. 
By reducing, ^vdy=.vydv — ydv. 

Dividing by vy^ and we obtain 

^=.dv-Jl^, 

y V 

By integrating 2.\og.y=v — log-v^-^' 

That is log.y-{-(\og.y'\-\og.v)=v-\-C, 

Or log.y+log.a:=-+ (7, the integral sought 

3. Integrate the diff^erential 



xdy — ydx = dx Jx ^ -f-y ^ 
If we place y=^vx, then 



xdy — vxdx = dx Jx ^ -\-v ^x^ . 



Dividing by x, dy — vdx=^dxj\ -\-v^ . ( 1 ) 

Because y=vx, dy=vdx-{-xdv, and dy — vdx=xdv. (2) 
Equating (1) and (2), we have 



xdv=dxJl-{-v'^. 
dv dx 



By integrating, we find 

log.(t;+^l+v2)=log.a;+log. C=log. Cx, (See Art. 81.) 



Passing to numbers, v-\-J\-^v^=:Cx 



Restoring the value of v ^-f- /l -f-^ = ^^• 

X \ «2 



336 INTEGRAL CALCULUS. 

Multiplying by ar, y+J^~+y^ = Cx^, 

4. Integrate the differential equation 

^^^ydx^y_^ (1) 

Let x=^vy. Then dx=.vdy-\-ydv. 

The values of x and dx substituted in (1), and reduced, will 
give 

, __^ dv 

By integrating, we find w= — arc(tan.=v.) (Art. 68.) 
Restoring the value of v, and w= — ^arcf tan.=- j-|-(7. 

5. tntegrate the differential' 

ax^dy — axydx 
du=^ "■" ~I~. 
(x^+yn^ 
Place a;=«;y. Then substituting the values^ of x and dx, we 
shall find after reduction 

avdv 

Whence w= — a T"" "T» 

*^ (1+V2)2 

Integrating the second member by formula B, (Art. 77,) we 
find w= — — Restoring the value of v, -, we obtain 



,— «y 



-j- (7, for the result. 



Jx^'\-y'' 

(Art. 112.) Differential equations may sometimes appear in 
the form 

dy+Pydx=Qdx, (1) 

in which P and g are functions of x. 

The object of this article is to show the integration of such 



PARTICULAR INTEGRALS. 337 

differentials. Equations of this kind being of the first degree, 
in respect to y and dy are sometimes called linear equations. 

Place y=zX, (2) 

X being some function of x, to be determined by circumstances, 
as we are about to explain. 
The differential of (2) is 

dy=zdX-\'Xdz. 
This value of di/ substituted in (1), produces 

zdX+X(ds+Pzdx)=Qdx. (3) 
Now Z* being arbitrary, we can so assume it that 

zdX=Qdx. (4) 

Then X(dz+Fzdx)=0. 

Whence X=0, or dz-\-Pzdx=^0. 

From the last — = — Pdx, 

z 

By integration log.0= — J Pdx. (5) 

But l=log.e. (6) 

By the multiplication of (5) and (6), we harVe 
log.s= — rPdx.\og.e=^log.e~/'^*^^. 
Passing to numbers, z=e—f^^^. (7) 

From (4) dX=^ = Q(eP^^ )dx (8) 

z 

By integration X=jQ(ef^^'')dx. (9) 

The values of z and X, (7), (8), substituted in (2), give 

the formula for the integral value of y. 

EXAMPLES. 

1 . Integrate the differential 

xydx adx 



dy- 



l+a;2 \J^x^' 



Here P=— __^_. Q= ^ 



338 INTEGRAL CALCULUS. 



Comparing this with (5), log z=\og. J \-\-x' . 

Or z=—^\-\-x'. (1) 

From (8) dX=^=-- -?-^ =- "^^ . 

By integration X=— -^_-+C. (Formula 5, Art. 77.) 



Vl+» 



But y=zX^=ax — CJ 1 -[-a;^ , the integral sought. 

2. Integrate the differential 

, __aydx hdx 



1 — ^a; 1 — X 



Here P=-— A_. Q=.J-, Pdx=. "^^ 



1 — X 1 — a; 1 — X 

J^Pdx=a log.( 1 — x). 
Hence \og,z—a\og.{\ — a:)=log.(l — xy. 

Whence 2=(1— »)*. (1) 

From (8) we have dX= ^^^ ^^"^ 



By integration X=— - ? 4- (7. (2) 

o(l — xY 

The product of (1) and (2) will give y for the first member, 
whence 

2/=— -+qi— a;)*. C^) 

a 

N. B. If this is truly the integral sought, its differential will 
produce the example. We will thus verify it. 

dy—aC(\-^xY-'^dx. 
Multiply both members of this equation by (1 — x)^ and 
( \—x)dy=^a C{\--xydx. 

(±=p^l^aC(l^xy. 
dx 



PARTICULAR INTEGRALS. 339 

From (3) we have 

ay-{-b=aC(\—x)\ 
Whence {\—x)dy={ay-\-h)dx. 

Or dy—^^'-=^^, the given differential 

1 — X 1 — X 

Thus we might verify the first example. 

3. Integrate the differential 

_^=_^cos. Q-\-2am.y. 
dQ a 

A _/^2 3mQ_j_ ^g&m .Q 4^mcos.^ 

j^ns. y-i.e -f--^-^^-^- T+To^^' 

This example is solved in the author's Operations. It is the 
last problem in that work. 

4. Integrate the differential 

dy-\-^axydx=hx^ dx. 

Here P=^2ax, and Q=bx\ jFdx=ax\ 

V/hence 2=e-"^ and dX=9^-=^l=bx'e''^dx. 

z e-" 

X=bjx'^e^^"dx. 
The integral of this last expression depends on a series, and 
therefore it can only be found approximately, and as the differ- 
ential applies to no particular problem or question in philosophy, 
we leave it thus : 

y^be-^""^ Jx^e^''^dx-\- C. 



340 INTEGRAL CALCULUS. 

Miscellaneous Hxaniples. 

1. Draw the line indicated by the equation 

2. Draw the line indicated by the equation 

3. Draw the lines indicated by the equations 

— y--\-x=0, and y-\-x=zO. 

4. Determine the angle formed by the intersection of the two 
lines indicated by the equations 

2y+4a;+l =0, y— 10a;-|-3=:0. 

Ans. The obtuse angle is 147° 43' 27"6. 
The acute angle is 32° 16' 34"4. 

5. Determine the angle formed by the intersection of the two 
lines whose equations are 

and find the co-ordinates of the point of intersection. 

Ans. The acute angle is 26° 34' 8", the obtuse angle is there- 
fore 153° 33' 62", and if we represent the required co- 
ordinates by x', y\ we shall find a;'=0.31, y'=7.9293. 

6. Describe the circle whose equation is 

That is, find the radius and the co-ordinates of the center. 
Ans. Let x\ y', represent the co-ordinates of the center, 
and R the radius, we shall find i?=6, a?'=3, y'= — 4. 

7. Describe the circle whose equation is 

ar2_j.y2_4^_4y_3^ 

8. Describe the curve whose equation is 

y=2a;3_5a;2+2. 

9. The hypotenuse of a right angled triangle is constant, 
but the perpendicular varies : what will be the corresponding va- 



MISCELLANEOUS EXAMPLES. 341 

riation of the other side, and what effect will be produced on the 
acute angles? 

10. What is the differential of n=2x+5x^ -\-b1 

Ans. du=(3-\-15x^ )dx. 

11. What is the differential of u=(a+Jxy'l 

Ans. du='<^±-^S2±. 
^Jx 



12. What is the differential coefficient of 



U: 



1-^x 



Ans. 



du__ 3 — X 

13. What is the differential coefficient of u=.aA-—^-. 

^^3+x^ 



Ans, ^= Jllz:^. 

^^ (3+x'yx' 

14. What is the first derived polynomial of the algebraic 
equation 

Ans. 3a;2— 34i»+54=0, 

15. What is the differential of tc:=a^-{-by'! 

Ans. du=^\og.adx-{-^\a^~\-by). 

16. What is the differential of u=xlog.x2 

Ans. du=('[-\-[og.x)dx. 

17. Differentiate u=\og.^]+J}z:^L)' 

dx 



Ans. duz 



xj\ — X- 

18. Find the arc whose logarithmic tangent varies three 
times as rapidly as the logarithmic cosine. 

Let the arc be represented by x. Then the problem requires 
that 3c?.(log. cos.a;)=cf.(log. tan.rc). 

A.ns. ;i'=35° 16' 9'^ 



342 INTEGRAL CALCULUS. 

19. Find the arc whose log. tangent varies five times as 
rapidly as the log, sine of the same arc. 

Here 5d.(\og sm,x)=d.(\og. tan.ar). 

Am. ar=63° 25' 52". 

20. Find the values of x which will render the function 

7/=ax^ — b^x^-\-C, 

a maximum or minimum. 

Ans. y is a minimum when x=0, and a maximum 

, 2b^ 

when iP= 

3a 

21. Divide the number 60 into two such parts that the square 
of one part diminished by 3 times the rectangle of the two parts 
shall be the greatest possible. 

Am. The parts are 22| and 37|. 

22. Find the greatest value of y corresponding to the equa- 

x^ / 



a — X 



Am. When y is greatest, x= — ■ , when least, x=0. 

At 

23. Required the sub-tangent of the curve wTiose equation is 

xy^-=ia^{a — x). 

Ans. -i(ff=f!_). 
a 

Qcix—~x ^ 

24. The sub -tangent to a curve is — , find the equa- 

a 

tion to that curve. 

Am. xy^=a^(a — x.) 
N. B. To resolve this, we place the general expression for a 

((^x\ 2 (ax x^ ) 
y—) equal to — — ^^ Z, and separate the 
dy/ a 

variables and integrate. 

25. What is the length of the longest straight inflexible pole 
that can be put up a chimney, when the hight from the floor to 
the mantel is =a, and the depth from the front to the back =6? 



Am. 



«V^+C-)'-^V'+G)*- 



MISCELLANEOUS EXAMPLES. 343 

26. Find the equation of the curve whose sub-normal is 4ar^y? 

i. 
Ans. y=3x^-{-b. 



27. The tangent of a certain curve is represented by y. /?jtl, 

^ X 

what is the equation of that curve, and what is the expression 
for its sub -normal? 

Am. The equation for the curve is y=2jx-\-b. 

The sub -normal is represented by i^. 

28. Required the area of a curve whose equation is xy^ :=a. 

Ans. The area is =2xy, 

29. Find the equation of a curve whose area is expressed by 
twice the ordinate. 

Ans. x=^\og.y-\-h. 
An equation in which x is the abscissa and y the ordinate. 

30. The sub-tangent of a curve is expressed by twice the 
rectangle of its co-ordinates. Find the equation of that curve. 

Ans. y=^\og.x-\-b. 

31. The expression for a tangent to a curve is — r . Find the 

X 

equation to that curve. 



Ans. We place y l\^^^ , the general expression for a 

Rv 

tangent equal to the given expression — ; and by reduction and 

x 

integration we find the curve to be a circle. 

32.. The sub-normal of a curve is |a;^-|-3a;-|-|^, find the equa- 
tion of the curve. 

Ans. y''=Zx^-\-Sx^+x-{-C. 

33. Find the equation of the curve whose area is expressed 
by two-thirds of the product of its co-ordinates. 

Ans. y'^z=:Cx, but we may assume C=2p, then we have 
y^z=z2px, the common parabola. 



344 INTEGRAL CALCULUS. 

34. Find the equation of the curre whose sub -tangent is 
equal to the rectangle of its ordinate and sub-normal, x being 
the abscissa, and y the ordinate, the curve commencing at the 
origin of the co-ordinates. 

Ans. y^=^x^, a cubical parabola. 

35. In latitude 40° north when the sun's declination is 10® 
north, what time in the day will the variation of the sun's alti- 
tude be the greatest possible? 

Ans. When the sun is due east or west. 

N. B, In spherical trigonometry, we learn that 

r> sin.^ — sm.L cos.i> 

cos.P= r--— n — ' 

cos.L sm.D 

an equation in which A= the sun's altitude, i/= the latitude, 
I) the sun's polar distance, and F the angular distance of the 
sun from the meridian. 

This problem requires us to find when dA shall be the greatest 
possible, L and D being constant quantities ; F will vary in con- 
sequence of the variation of A. 

' njn COS,. Ad A /,x 

— mn.FdF= — -. ( 1 ) 

cos.Zsm.Z/ 

But cos.^ : sin.P : : sin.Z> : sin.Z, 

Z being the sun's azumuth. Whence cos.^= — '—, 1— . 

"^ sin.^ 

This placed in (1), and reduced, we find 

dA=—dP. cos.Z sin.Z. (2) 

That is, dA is the variation of altitude for any small interval 
of time corresponding to dP, (the variation of the angle P being 
uniform,) therefore, as — dFj and cos.L are constant quan- 
tities, dA is greatest when (sin.-2) is greatest, or when the center 
of the sun is due east or west. 

By means of the right angled spherical triangle we find in 
Lat. 40° north, when the sun's declination is 10° north, the sun 
must be due east 5h. 11m. 28s. before it comes to the meridian, 
and the same interval after meridian would bring it due west, 
provided the declination did not change during the interval. 



MISCELLANEOUS EXAMPLES. 345 

36. In the last example we required the time of day whca 
the variation of the sun's altitude is zero. 

Atis. This is answered by placing g?^=0 in (2) of last 
example. Then the second member of that equa 
tion is 0, which makes Z=0 or dF=0. Then the 
sun is on the meridian, or it is apparent noon. 

37. The area of a curve is represented by x'^y, what is the 
sub-tangent to that curve? 

Ans. 

1— 2a; 

38. The sub-normal of a curve is ^, what is the equation 

X 

of the curve? 

Ans. y^ =2a2 log.ar. 

39. A curve is expressed by fl-l-z=zrdx, what curve is 
it, or what is the equation of the curve? 



Ans. y= J2rx — x^ , showing that it is the equation of 
the circle, the origin being on the curve. 

40. The base of a right angled triangle is a, and the perpen- 
dicular X, and hypotenuse y ; x and y are variable : what relation 
must exist between x, y, and a, when the variation of ic is w times 
that of y ? 

Ans. y=^nx, and x= — 



Jn^—\ 

41. Taking a triangle as designated in the preceding propo- 
sition, a variation of the perpendicular and hypotenuse will 
necessarily involve a variation in the acute angles. Determine 
that variation. 

Ans. The acute angles will vary by a quantity whose 

sine or tangent is measured by , in words, 

The sine or tangent is equal to the base multiplied by the variation 
of the perpendicular, and that product divided by the square of the 
hypotenuse: 



346 II^TEGRAL CALCULUS. 

Ex. — The base of a right angled triangle is 80, the perpen- 
dicular 60, and the hypotenuse 100 feet. 

If the perpendicular be increased or diminished jV of a foot, 
what will be the corresponding variations of the acute angles? 

Ans. The nat. sine or tan. is .?5jJZ_=0008, 2' 45" 

10000 

log. .0008 —4.903090 

Add 10 

log. sine 6.903090 (See Robin- 
sub, log. of 1" ...4.685575 son's Geom. 

log. of 165" 2.217515 page 161.) 



42. Integrate the equation -^=B — A cot.a;. 

dx 



dx 

Ans. y=- C-\'Bx—'A log. sin.a;. 



43. The hypotenuse of a right angled triangle is given. Re- 
quired its dimensions when tlie perpendicular added to twice 
the base is a maximum. 

Ans. If h represent the hypotenuse, hj\ is the 
perpendicular, ^hj\ is the base. 

3. 

44. The area of a curve is represented by |5^^, x and y being 
the co-ordinates; the curve commencing at the origin. What is 
its sub -normal? 

25a;* 

Ans. The equation of the curve is y=. , and the 

36 

25 
value of its sub -normal is — xy. 

45. What is the sun's longitude when its variation in longi- 
tude is 10 times its variation in declination? 

N. B. Let D represent the sun's declination, L its longitude, 
E the obliquity of the ecliptic. Then the fundamental equation 
is sin.D=sin.^sin.Z. (i) (Radius unity.) 

By diflferentiation c,OQ.DdD=^B,m.£J QOB.LdL. (2) 

The condition requires dL=:lOdD, 



MISCELLANEOUS EXAMPLES. 347 

This substituted in (2) and reduced, produces 

cos.i>=10sin.^cos.Z. (3) 

By squaring (1) and (3), and adding them, observing that 
ein.2i>-j-cos.2i)=l, and still further reducing, we shall have 

cos.^X= , or cos.//= — ■ 

when n represents the ratio of the variation expressed generally. 

And this is the answer in general terms. 

Ans. The sun's longitude is 76° 37' 12" from the equinoxes, 
that is to say, Ion. 76° 37' 12", 103° 22' 48", 
256° 37' 12", and longitude 283° 22' 48". 

46. In latitude 42° north, when the sun's declination was 12° 
north, the shadow of a perpendicular post, 10 feet high, extend- 
ed 22 feet horizontally, it being in the forenoon . What was the 
time of day, and what time must elapse for the shadow to con- 
tract ,^^ of a foot? The semi-diameter of the sun being 1 5' 54". 

Let ^= the altitude of the sun at the time the shadow ex- 
tended 22 feet. Then the tangent of the apparent altitude of the 
upper limb is found by the following proportion : 
R : tsin.A : : 22 : 10 tan.^=log.l 1.000000— log. 1.342423= 

log. tan.^=9.657577=24° 26' 39" 
When the shadow was 21.7 feet, the alt. was =24° 44' 30" 



The difference of these altitudes is 17' 51"=107r', 

which we take for the differential of the first altitude. 

The altitudes computed from a shadow correspond to the upper 
limb of the sun, — therefore to obtain the true altitude of the 
sun's center at the same time, we must subtract the sun's semi- 
diameter and the refraction. 

In this case the sun's semi-diameter=15' 54", and the refrac- 
tion 2' 8", both subtractive. Hence, from 24° 26' 39" Ave take 
18' 2", and we have 24° 8' 37" for the sun's altitude when the 
shadow of 10 feet perpendicular extended 22 feet horizontally. 

Let Z= the latitude of the observer, and D= the sun's polar 
distance, then with the true altitude of the sun's center, we find 
its meridian distance P=68° 12' 40", and the time from appa- 
rent noon is 4h 32m 51s, or it is 7h 27m 9s apparent time A. M. 
(See Robinson's Geometry, page 211.) 



348 INTEGRAL CALCULUS. 

Solar time is deduced from the spherical equation 

p_sin.^ — sin.L COS. JD (See Robinson's 
cos.^ ^sX^i^i9 ' (>eom. p. 209.) 

in which L and D are constant quantities, and P varies in con- 
sequence of the variation of A, therefore by taking the differ- 
ential, we shall have 

cos.Zsin.i) 

Or <^P=— . ^Q^'^^^ , (radius unity.) 

sm.Pcos.Zsm.i> 

The minus sign indicates that P decreases while A increases, 
which is true whatever be the time of day. 

To find the value of dP, we have ^=24° 8' 38", c?^=1071", 
Z;=42°, Z)=78°, and P=68° 12' 40". 

cos.^ 24° 8' 38" (radius 1) —1.960243 

dA 1071" 3.029789 

2.990032 

sin. 68° 12' 40" —1.967810) 

cos. 42° — 1.871073> 

sin. 78° , —1.990404) 

—^'829287 

<;P=1448" 3.160745 

Fifteen seconds of arc correspond to one second oi time, there- 
fore 1448" corresponds to I minute 36^ seconds, and in this 
interval the shadow will contract three-tenths of a foot, 

If to 7h 27m 9s we add Im 36s, we shall have 7h 28m 45s 
for the mean time when the shadow extended 21.7 feet. 

If the second altitude be corrected, and the time correspond- 
mg be computed, the result will be 7h 28m 48s, a result within 
three seconds of the differential method, but the differential 
method is the most accurate for small differences. 

47. The logarithmic differential of the sine of an arc is six 
times the logarithmic differential of the cosine of the same arc? 
What is the arc? 

Ans. 22° 12' nearly. 



H 258 83 



\ 



0' 



^""^ 




o > 









^-./' 




^ ^ 
%^\^ 





^^^^ 










^^;--'\/\ V"^'"/' V'-T^-\#' 





V * o. -> 









^T.',< 








-oK 




'^ 












^ ^ 

^^\^ 









. f^*" , O « G ^ <ii 








ANT * A->^'r> -" ^:^MIlillllP^ « 

^^ ^' , ^-^ "•* aV" <- -^^^ A<^ 







.HO 



.:%/ 



o v^ :^ 



'>^ 







